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Answers: 1. Analytic def. 6. Demonstrative def. 11. Synonymous def. 2. Def. by sub-class 7. Operational def. 12. Demonstrative def. 3. Synonymous def. 8. Analytical def. 13. Analytical def. 4. Enumerative def. 9. Enumerative def. 14. Operational def. 5. Etymological def. 10. Def. by sub-class 15. Analytical def. # Analyses Closely related to analytical definitions of words are analyses of concepts or kinds of things. We might want to know what the word ‘Justice’ means, or we might want to know what Justice is. They are related questions, to be sure, and if we know the answer to one, we may be on the road to knowing the answer to the other. The Greek philosopher Plato (428-347 BC) wrote dialogues featuring his real-life teacher Socrates (469-399 BC) as the main protagonist. Socrates wandered about Athens in real life and in Plato’s dialogues asking for accurate, detailed analyses of moral virtues like courage, wisdom, and justice. He wanted to understand what these virtues were —in part —so that he could more readily adopt them into his own character. Socrates rarely found anyone who could provide a good analysis, and philosophy students around the world today cut their teeth on these dialogues to fine-tune their analytic skills. When Socrates appeared to be asking for definitions for words, he was actually looking for a cognitive understanding of the necessary and sufficient conditions for something being the kind of thing it was. If he had asked about the nature of a trian gle, he’d be searching for something like “A triangle is an enclosed geometric figure with three sides.” If he had asked for the definition of ‘triangle’ (i.e., the word), he’d likely have been happy with “‘Triangle’ means an enclosed geometric figure with three sides.” The responses are similar, but it’s a somewhat
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different challenge to determine the essential nature of a thing as opposed to what is accurately meant by the word used to refer to that thing. Justice is what we want for our society; ‘Justice’ is a word with seven letters. We’ve just now referred to the essential nature of a thing. That needs a little explanation. A thing’s essential nature— in the present context —refers to the character traits or properties that thing must have to be the kind of thing it is. That sounds more complicated than it is. Think of a square. What traits must it have to be a square? For one, it must be an enclosed geometric shape. Having an enclosed geometric shape is true of some things —like rectangles and circles —but false of others —like desires for cheeseburgers, the possibility Bob might fall in love with Mary, and Justice. What’s true of a thing can be said to be one of its properties. You can’t be a square unless you’re an enclosed geometric shape. Thus having an enclosed geometric shape is an essential property of a square. But you’ve also got to have four equal sides. And have four right angles! Each of those traits is necessary for you to be a square. As it is, that list of conjoined 190 character traits —enclosed geometric figure composed of four equal sides and having four right angles —is sufficient for you to be a square. If you have all of those traits (or properties), then that’s enough to guarantee your squareness. A complete and accurate analysis of a square— or squareness —will thus provide the traits that are necessary and sufficient for squareness. Objects in our day-to-day, ordinary world will also have accidental character traits. These traits are characteristics that the thing has but
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does not need to be the kind of thing it is. A table —as a table —will have a flat, raised surface; that will be an essential trait of a table. But the table might also be colored blue. Blueness is not essential to the table, as we can paint the object completely red, and it will still be a table. Blueness or redness are thus accidental traits of a table. The concept of a condition is more general than that of a cause , as all spatio-temporal causes are conditions of effects, but not all conditions are spatio-temporal causes. The presence of gasoline is a necessary cause for a standard automobile to run, and we can also say more generally that the presence of gasoline is a condition that must obtain for the car to run. However, a fistful of an odd number of coins is a necessary condition for the fist to hold exactly three coins, but being odd in number is not exactly a cause of one’s holding three coins. Also, having a flat surface will be a condition that must obtain for an object to be a table, yet it will sound peculiar to refer to having a flat-surface as a cause of a thing being a table. In an analysis of a thing or concept, we are looking for conditions that obtain that make the thing or concept what it is. We do not want to appeal to accidental conditions, for a thing does not need those traits to be the kind of thing we’re analyzing. We want to refer to all and only the essential character traits, that is, to the necessary and sufficient conditions for an X being an X. And this is rarely easy, especially with philosophically interesting and scientifically complex concepts.
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Heck, it’s even a challenge to analyze what a chair is. Let’s try. Attempt #1: A chair is something like that over there [as we point to a chair]. But, for all I know, what you’re pointing to is wooden th ings, or things painted brown, or things owned by humans. To what exactly are you intending to draw my attention? I still don’t know what a chair is. Attempt #2: A chair is something like a barber’s chair, a beach chair, a director’s chair, a recliner, or a child’s high chair. But none of this tells me what a chair is . I continue to be ignorant about the nature of chairs. Attempt #3: A chair is something that is accurately referred to by the word ‘chair’. You’re kidding, right? Attempt #4: A chair is something we can sit on. But that includes pillows and the floor. That may be a necessary condition for a thing being a chair, but it’s not sufficient. Your fourth attempt doesn’t help much. Attempt #5: A chair is a raised seat we can sit on. But that includes stools, which are not chairs. Arg! Try again. 191 Attempt #6: A chair is a raised seat that we can sit on, with a back. But that includes couches, doesn’t it? And couches are distinct from chairs. Attempt #7: A chair is a raised seat with a back upon which only one person can sit. Why couldn’t that describe a bar stool with a back? Should I be going to Wiki for a definition? Huh? Attempt #8: A chair is a seat with a back, where the seat is raised to approximately knee level of an average adult so that he or she can sit on it. But that excludes two-inch chairs for doll houses,
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and small though they may be, they’re still chairs. Yes? Egad! I thought I was asking a simple question! Let’s give up and leave this task to chair experts. We’re not saying it’s impossible to analyze the nature of a chair, but it’s often more challenging than it looks. No wonder the people Socrates conversed with long ago had such trouble analyzing more complex things like Knowledge, Justice, Goodness, and Courage. And yes, there’s vocabulary distinct to analyses! Just like a definition is made up of a definiendum and a definiens, so too does an analysis consist of an analysandum (the concept to be analyzed) and an analysans (the words doing the analysis of the analysandum. To summarize, we can say that an accurate and complete analysis of a concept or thing will provide all and only the essential properties of that thing. In other words, such an analysis will provide the necessary and sufficient conditions for something being the kind of thing it is. If I am analyzing what it is to be a chair, then I will provide all the essential character traits of chairs (as chairs), and nothing else. And that’s not always easy; and according to some philosophers, it’s sometimes impossible . But let’s go as far as we can with analysis. Let’s not give up simply because the going gets rough. Clarity is almost always a good thing. # **Practice Problems: Essential and Accidental Properties Are each of the properties below essential or accidental to the kind of thing in question? 1. Three-sidedness is a property of triangles. 2. Having a four-inch-long hypotenuse is a property of triangles. 3. A yardstick must have measuring marks. 4. Water possesses hydrogen. 5. Senator Sunny Shine gardens clothes-free whenever she can. 6. Pastor Bustle is a Central Baptist
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minister. 7. Dictionaries contain definitions of words. 8. Oil-based paint has oil in it. 9. This pencil is made of yellow-colored wood. 10. This pencil has the property of being usable for drawing. 11. The line making up this circle has the property consisting of points. 12. This circle has the property of being one foot in diameter. 13. All dogs have the property of being animals 192 14. No cats are fish. 15. Some birds are green. Answers: 1. Essential 6. Accidental 11. Essential 2. Accidental 7. Essential 12. Accidental 3. Essential 8. Essential 13. Essential 4. Essential 9. Accidental 14. Essential 5. Accidental 10. Essential 15. Accidental # **Thinking Problems: Essential and Accidental Properties 1. What are the essential properties of the Supremely Perfect Being (i.e., God)? 2. What are the essential properties to being a human? 3. What are the essential properties of a just social system? Answers: 1. Traditional theism includes omnipotence, omniscience, omnibenevolence, and eternality. Philosophical theology is filled with discussions about this topic. The topic can be of interest whether you are a theist or not. Even an atheist might wish to experiment with the hypothetical question, “If God exists, what might be an accidental character of such a being?” 2. The Aristotelian tradition points to the use of reason; John Stuart Mill —at least in his On Liberty —seems to favor free will as an essential feature of humanity; the Confucian tradition focuses on our social nature and capacity to form key relationships. If you can come up with a better answer, publish it, get rich, and pave the way for a truly just political environment. 3. There is no way this text is even going to try to answer this one. That’s what Social Philosophy and Political Philosophy classes are for. #
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Mistakes in Definitions and Analyses The problems we had above trying to give an accurate, full analysis of a chair or definition of ‘chair’ points to a number of common mistakes. Let’s look at some in detail. Obviously, any definition or analysis needs to be accurate. If we define the word ‘T -shirt’ to mean “a large striped cat living in India,” we’d be mistaken, wrong, and muddle -headed. Some people simply misunderstand or are ignorant of the meaning of certain words, or cannot successfully and accurately describe the essential character traits of a thing. What we are looking for here are basic traits any good, useful definition should possess, when they are intended to convey cognitively the accurate analysis of a thing or the meaning of a word. A good analysis of a kind of object and a good analytical definition of a word should not be the following: Merely extensional Too broad Too narrow Circular Negative Unclear 193 Merely Extensional Since extensional definitions only provide examples, they are unlikely to be a satisfactory definiens or analysans. Socrates was constantly running into this problem in Plato’s dialogues. He’d want an informative analysis of a moral virtue, and usually the first thing he’d hear was an example of that virtue. In the dialogue Euthyphro , he wanted to know what piety was, but was initially told that it’s prosecuting wrongdoers. Even if true, this hardly gave Socrates an understanding of the essential nature of the virtue. Although extensional definitions may provide useful illustrations for more informative intensional definitions or analyses, they are our first mistake: When offering a definiens or analysans, avoid being merely extensional . For example: * ‘Rock -n-Roll band’ means groups like the Rolling Stones, AC/DC, and Black Sabbath. * ‘Rock’ means materials that are igneous, sedimentary,
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or metamorphic. * Courage is being willing to obey orders on the battlefield when under heavy fire. * Honesty is saying “No” to one’s boss when she asks you to lie to a customer. Too Broad A common mistake in analyses and definitions, and one that is often a challenge to avoid is being too broad . Here, the analysans or definiens covers more than i t should; it’s too broad in scope; it covers what it’s supposed to, but more. For instance, “A tiger is a large, fierce cat.” As an analysis of a tiger, “large, fierce cat” covers all tigers, but also includes lions, cheetahs, and leopards. Better would be “A tiger is a large, fierce, striped cat.” Such an analysis has the advantage of not including lions, cheetahs, and leopards. If a proffered analysis is too broad, then what we can do to relieve our poor benighted friend of his befuddlement is to point out examples of things that are clearly included in the concept being analyzed, but which sadly fall outside his analysans. “A truck is a vehicle,” our confused friend might say by way of a brief analysis of his favored means of transportation. In the spirit of bon ami , we might reply, “Your analysis falls just shy of coherent, for cars, motorbikes, and hovercrafts are a vehicles, too, yet only the delusional perceive them as trucks.” Other analyses that are too broad include the following: * A ketch is a sailing vessel. * A boy is a young human. * A pencil is an instrument used for writing or drawing. * A chainsaw is a tool used for cutting wood. Definitions that are too broad include, analogously, the following: * ‘Ketch’ means sailing vessel. * ‘Boy’ me ans a young human. * ‘Pencil’
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refers to an instrument used for writing or drawing. * ‘Chainsaw’ means a tool used for cutting wood. 194 Too Narrow The flip side of being too broad is being too narrow . Here the definiens or analysans covers too little; the an alysandum or definiendum refers to things that are not included. For instance, “A tiger is a large, fierce, striped cat presently living in India.” The most obvious problem here is that there are tigers that presently live outside of India, perhaps in zoos. The analysans here is too finely focused; it’s not broad enough; it’s too narrow. Other examples of analyses and definitions illustrating this problem include the following: * An abode is a three-bedroom house with two baths. * A birdhouse is a dwelling for wrens. * A musical instrument is something producing sound for a piece played in an orchestra. * ‘School’ means an environment in which history is studied formally in classroom settings. * ‘Soccer’ refers to a competitive sport played with a ball. * ‘Battleship’ means a large, sea -going vessel. Circular For obvious reasons, we don’t want to assume people understand the word or concept being defined or analyzed when we offer our definition or analysis. If we define a word by using that same word —or a near variant of it —we are not helping folks much. Examples of egregiously circular definitions include: * ‘Musical instrument’ is an instrument used to play music. * ‘Loneliness’ is the state of being lonely. * ‘Happy’ means not being unhappy. * ‘Wood cutter ’s ax’ refers to an ax used by a wood cutter. Sometimes definitions and analyses can be circular in more subtle ways. Imagine a person who for whatever reason does not know what money is. You get air-lifted to her isolated
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village, strike up a conversat ion, and use the word ‘money’. She asks, “What do you mean by the word ‘money’?” As a committed advocate of capitalism, you welcome the chance to afford this childlike soul insight into your society’s highest value. “The word ‘money’ means the thing made b y a mint.” The problem is that if this otherwise fulfilled woman does not know what money is, she can hardly be expected to know what a mint is. You need to know what money is in order to make any sense of an institution whose primary function is to make money. The same problem can occur in analyses. When Socrates was hunting around for an analysis of piety in Plato’s Euthyphro , one response was “Piety is that which pleases the gods.” Since that did not really tell Socrates what piety is , he asked what it was that pleases the gods. It turned out to be piety. So, piety is that which is pious. Sigh. Negative 195 A simple problem to recognize in definitions and analyses is being negative instead of affirmative. That is, when we can, we want to explain what a word means rather than what it does not mean. When possible, we want to provide the necessary and sufficient conditions for a concept, and not say what they are not. To do the latter in each case fails to explain the meaning of the word or the nature of the thing being analyzed. Note how the following definitions fail to explain what each word means: * ‘Tack hammer’ is a tool that is not used for setting screws. * ‘Sloop’ is a boat that is not a ketch. * ‘Harmony’ is not a state of discord. In each case, we could have done better by
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trying to say what the word does mean: * ‘Tack hammer’ is a tool used for pounding small nails, brads, or tacks. * ‘Sloop’ is a one -masted sailing vessel. * ‘Harmony’ is a state of concord. Some words require a negative definiti on, and in such cases, there’s nothing to be done than to define them in negative terms. For example: * ‘Bald’ means having no hair. * ‘Darkness’ means absence of light. * ‘Vacuum’ refers to an absence of air. The same mistake can be found in unsatisfactory analyses: * Corporeal substances are those having no thought, will, or consciousness. * Being good is to avoid acting in selfish ways. As with definitions, some concepts require a negative analysis. The concept of darkness, for instance, will be analyzed in terms of absence of light. Still, when we can, we should try to analyze a concept in terms of what it is, rather than in what it is not. Unclear We place here at the end a hodgepodge list of problems that boil down to making the definition or analysis unclear. This is more often than not due to poor writing skills, and less often to a lack of commitment to presenting a clear explanation of a word or concept to help convey cognitive meaning. A definition or analysis may be unclear due to poor grammar, or by being vague, figurative, emotive, or needlessly complex. We’ll look briefly at each, focusing on definitions, but each problem can apply to analyses, too. It takes careful writing to select words with precision, and poor grammar can get in the way. Students in a Critical Thinking class should know that arguments contain premises and conclusion. But consider this definition: “The word ‘argument’ means where you have a set of 196 statements, one
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or more of which are premises, and one other of which is the conclusion that follows from the premises .” The problem here is with the use of ‘where’. An argument is not a place, so “where” makes no sense here. Better would be this: “The word ‘argument’ refers to a set of statements, one or more of which are premises, and one other of which is the conclusion that follows from the premises.” Here are two more examples of poor grammar getting in the way of a definition being successful: * A ‘statement’ is when a sentence is true or false. [Statements are not a tim e.] * Bill spoke about the word ‘square’. Meaning a four -sided enclosed geometric figure with four right angles. [Fragmented sentences convey no meaning.] Another problem with grammar can reflect a definition’s unwanted ambiguity. An ambiguous definiens has two clear meanings, and thus the intended meaning of the definiendum is unclear. For instance, * In the game of chess, a player is said to be “ mated ” when one player moves pieces so that the other player ’s king is in check but cannot move it legally. Here, we cannot tell who is mated. Is it the player who can still move his or her queen, or the other player who cannot do so? A better definition that avoids this problem would be, “In the game of chess, a player is said to be ‘mated’ when his or her king is in check but cannot move it legally.” If a keyword in our definiens is vague , then we’ll fail to convey in an informative fashion what the word means that we are trying to define. Recall that a word is vague if even with a general understanding of it s meaning,
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we still don’t know what the word is supposed to mean in the present context. For instance: * ‘Bonsai’ means a small tree planted in a tray. [The word ‘small’ is vague here. Better would be “‘Bonsai’ refers to a tree less than six feet in height planted in a tray.”] * ‘Desert’ refers to a dry place. [‘Dry’ is vague here; it would be better to refer to maximal inches of annual rainfall.] We also want to avoid merely figurative language. Here we refer to poetic, whimsical, or metaphoric forms of expression. It may make for entertaining writing or discourse, but it rarely conveys cognitive meaning. In other words, it can be fun —and even funny —but it does not tell us what the word actually means. For instance: * ‘Puritan’ refers to one who fears th at someone, somewhere is having fun. * ‘Love’: a rose with gentle blooms and painful thorns. * ‘Chess’ refers to the game of kings. Since we usually want our definitions to convey cognitively the meaning we intend, we also want to avoid using emotive or affective language. Such language may psychologically persuade 197 and stir feelings —and there may be nothing wrong with that —but a roiling of one’s emotions is not the same thing as getting across clearly what we mean by a word or concept. Note the emotionally charged words used in each definition below. They are intended to sway readers to a point of view, rather than to educate them regarding our intended meaning of a phrase. * ‘Free speech’ refers to the natural right of every citizen to speak his or her mi nd, openly and unafraid of tyrannical censure, on matters great and small, for the health of a thriving democracy. * ‘Free speech’ is what racists,
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sexists, and homophobes dishonestly employ to spew their vile hatred upon those unjustly marginalized with less political power. Finally, a definition can be unclear because it is needlessly obscure . Again, if the purpose of our definition (or analysis) is to explain the meaning of a word (or to provide its necessary and sufficient conditions), then speaking in a pointlessly obscure or overly complex style will get in the way of informing our listeners. Of course, some words and concepts are quite complex, and demand a complex definition or analysis. The word ‘dynatron’ probably needs something as complex as “a vacuum tube in which the secondary emission of electrons from the plate results in a decrease in the plate current as the plate voltage increases.” If there is a simple and straightforward way of defining a word accurately, however, then w e’ll want to proc eed in that fashion. There is no good reason to use fancy, bizarre language when normal, conversational wording will do. We should thus eschew obscurantism. Those offering the following definitions should be slapped upside the head: * ‘Dustbuster’ means a handheld, mechanical, motorize atmospheric pressure gradient creator for removal of particulate matter. * ‘Soap’ means a saponified glyceride intended as a sanitizing or emulsifying agent. # **Thinking Problems: Mistakes in Definitions and Analyses Read Pl ato’s delightfully short Euthyphro or Book I of Republic , and look for various bad definitions and analyses of piety and justice respectively. The works can be found online for free and in many good print editions. # **Practice Problems: Mistakes in Definitions and Analyses What is the single most obvious problem in each of the following definitions and analyses? (A) Too broad, (B) Too narrow, (C) Circular, (D) Merely extensional, (E) Negative, (F) Unclear. Some may be guilty of
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more than one problem, but select the most obvious mistake. 1. A trout is a fish. 2. ‘Happiness’ refers to the state of being happy. 3. ‘Gourmet chef’ does not refer merely to fry cooks. 4. A university is something like Harvard, Princeton, or Yale. 5. A fish is something with gills that swims in the Atlantic Ocean. 6. A test is when one demonstrates mastery of some knowledge or skill. 7. Justice is paying one’s debts and telling the truth. 8. A camel is a ship of the desert. 9. A guitar is a stringed musical instrument. 198 10. Intelligence is what intelligent people have. 11. ‘Telephone’ means a device used for communication. 12. Birds are things like parrots, eagles, and doves. 13. Football is a senseless sport watched by couch potatoes wishing to escape their sedentary lives. 14. A skeptic is a non-believer. 15. ‘Sibling’ means sister. 16. Socialist health care is bank-breaking, unwarranted give-away to loafers. 17. ‘ Deodorant ’ means a preparation for camouflaging the malodorous secretions of the apocrine sudoriferous glands. 18. Architecture is frozen music. 19. A musician is someone who plays music. 20. A bonsai is a small tree. 21. A statue is something like that [as he points to Michelangelo’s David ]. 22. ‘Democracy’ refers to the self -affirming right of a people to chart their own destiny to greatness. 23. A ‘sound’ argument is where the premises would guarantee the conclusion and the premises are true. 24. A tiger is a carnivorous animal. 25. A bird is a feathered animal from South America. 26. A devout Christian is not agnostic. 27. Investing money in the stock market is legalized gambling. 28. ‘Street cleaner’ refers to a public thoroughfare sanitation engineer. 29. ‘Sandwich’ refers to two pieces of bread holding slices of
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roast beef and cheddar cheese. 30. ‘Italian food’ means it ems like spaghetti, lasagna, and tortellini. Answers: 1. Too broad 16. Unclear (emotive) 2. Circular 17. Unclear (needlessly complex) 3. Negative 18. Unclear (figurative) 4. Merely extensional 19. Circular 5. Too narrow 20. Unclear (vague) 6. Unclear (grammar; a test is not a time) 21. Merely extensional 7. Merely extensional 22. Unclear (emotive) 8. Unclear (figurative) 23. Unclear (grammar; an argument is not a place) 9. Too broad 24. Too broad 10. Circular 25. Too narrow 11. Too broad 26. Negative 12. Merely extensional 27. Unclear (figurative) 13. Unclear (emotive) 28. Unclear (needlessly complex) 14. Negative 29. Too narrow 15. Too narrow 30. Merely extensional 199 # Chapter 14: Probability # Probability Theory Inductive arguments purport to show that a conclusion is probably true given the truth of the premises. Strength of inductive arguments —as students will recall —comes in degrees. Some inductive arguments are strong, some are really strong, and some are really really strong. Weakness comes in degrees, too. Probability theory attempts to quantify probability, since the difference between really strong and really really strong is shy of transparent. Probability theory thus attempts to make the degree of probability clearer and more precise. There is more than one way to understand probability, and different probabilistic claims will mean somewhat different things depending on what kind of probability is intended. You can even at this initial stage likely sense the difference between the following statements: * It is probable that a six-sided die rolled honestly will turn up with a number greater than one. * The probability of a 17-year-old male getting in an auto accident within ten years of driving is greater than that of a 17-year-old female. * It is probable that Dan and Sue will get
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married this year. To determine the probability of these events requires our using different approaches to probability. We’ll thus examine three theories of probability: the Classical Theory , the Relative Frequency Theory , and the Subjectivist Theory . Each is distinctly suited for specific kinds of circumstances, and each can thus help us make sense of the meaning behind “It is probable that X.” First, let’s look at how proba bility is quantified. An absolutely guaranteed event has a probability of 1; an event that cannot possibly happen has a probability of 0. Thus probability can be expressed in terms of decimal numbers between and including 0.0 and 1.0. A probability of 0.5 indicates that there is a half chance of the event occurring. A probability of 0.25 indicates a one fourth chance. We can also see that probability can be expressed in terms of fractions. An honest flip of a coin will result in an even chance of either heads or tails, so there is a 1/2 or 0.5 probability of getting heads. There are six sides to a normal die, so an honest toss will give us a 1/6 or 0.166 probability of getting, say, a 2. We can also express probability in terms of a percentage. If there is a 1/4 chance of winning a bet, then we stand a 25% chance of winning that bet; if we stand a 0.17 probability of losing a bet, then we stand a 17% probability of losing that bet (to covert from a decimal number to a percentage, just shift the decimal two places to the right). Finally, we can think of probability in terms of odds. While probability is often thought of as a fraction, odds are thought of as a ratio. If we have fifty/fifty chances, that gives
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us a probability of 1/2. That converts to odds of 1:1. That is, there is one chance of winning (the left number in the ratio) to one chance of losing (the right number). A perfectly ridiculous way to convert from probability to odds, or from odds to probability, is to chant one easy mantra: “top -left, top-left, 200 top-left….” Notice that fractions (the probability) have top and bottom numbers (i.e., the numerator and denominator respectively), while ratios (the odds) have left and right numbers. The top number of the fraction and the left number of the ratio w ill be the same. So, let’s say you know that the probability of an event happening is 1/3. The odds will be 1:x. You don’t know what x is yet, but don’t sweat it; you’re half way there. Now think of the : as if it’s a plus sign. The two odds numbers toget her add up to the bottom number of the fraction. Soooooo… 1+x=3. That means the x is 2. BINGO! If the probability of an event is 1/3, then the odds are 1:2. You have one chance of winning against two chances of losing, which is exactly what you’d expect with a 1/3 probability of winning. For those who appreciate clarity and precision, the following formula expresses how odds work: Odds(A) = f:u In English, this says, “The Odds of event A happening can be stated in terms of a ratio of favorable outcomes ( “winners”) to unfavorable outcomes (“losers”). The odds-probability shift can work from odds to probability just as easily. Imagine you’ve got 2:3 odds of winning a game. What would be the probability? Think “top -left, top-left- top- left….” The left number from the odds is 2, so the top of the fraction is 2. So
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far you’ve got 2/x. The x will be the two odds numbers added together: 2+3=5. So the bottom number of the fraction is 5, making the probability 2/5! 2:3 2/5 The top-left numbers are the same. 2:3 2/5 2+3=5 The two odds numbers add up to the bottom fraction number. A cleaner, more precise formula might look like this: Odds(A) = x:y is equivalent to P(A) = x/(x+y) Here are some examples of equivalent quantified probabilities: Fraction Decimal number Percentage Odds 0/6 0.0 0% 0:6 1/2 0.5 50% 1:1 1/4 0.25 25% 1:3 1/3 0.33 33% 1:2 1/5 0.2 20% 1:4 1/8 0.125 12.5% 1:7 3/78 0.038 3.8% 3:75 17/91 0.187 18.7% 17:74 1/1 1.0 100% 1:0 # **Practice Problems: Quantifying Probability Provide the requested probability or odds. 201 1. What is the probability of 4/5 in terms of a decimal number? 2. What is the probability of 4/5 in terms of a percentage? 3. What are the odds of an event with a probability of 4/5? 4. What is the probability of 0.9 in terms of a fraction? 5. What is the probability of 0.9 in terms of a percentage? 6. What are the odds of an event with a probability of 0.9? 7. What is the probability of 75% in terms of a decimal number? 8. What is the probability of 75% in terms of a fraction? 9. What are the odds of an event with a probability of 75%? 10. What is the probability if the odds are 2:7? 11. What is the probability if the odds are 5:1? 12. What is the probability if the odds are 23:90? 13. What is the probability if the odds are 90:23? 14. What are the odds if the probability is 5/6? 15. What are the odds if the probability
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is 94/113? 16. What are the odds if the probability is 3/19? 17. What are the odds if the probability is 2/8? 18. What is the probability if the odds are 2:6? Answers: 1. 0.8 7. 0.75 13. 90/113 2. 80% 8. 3/4 14. 5:1 3. 4:1 9. 3:1 15. 94:19 4. 9/10 10. 2/9 16. 3:16 5. 90% 11. 5/6 17. 2:6 = 1:3 6. 9:1 12. 23/113 18. 2/8 = 1/4 # The Classical Theory The Classical Theory of probability was developed in the 17 th century to analyze the probabilities involved in games of chance. Two conditions must obtain for us to use this approach to determining how probable an event will be. First, the total number of possible outcomes must be known. Second, each outcome must have an equal chance. For instance, an honest toss of a coin has only two realistically possible outcomes: heads or tails. It is logically possible that it could land and remain on its edge, but for all intents and purposes, that logical possibility is not —in this sense —possible. Moreover, if it is an honest toss, then there is just as much chance that it will turn up heads as tails. The same is true of a roll of an honest six-sided die. There are e xactly six possibilities (it won’t land and remain on an edge or corner): 1, 2, 3, 4, 5, and 6; and each side has an equal chance of appearing. If the die is “loaded” or weighted, it is not an honest toss, and the chances of any one number coming up are not equal to the others. The Classical Theory would then not give an accurate analysis of the probability of any one number coming up. Or, if we did not know how 202
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many sides the die had (it might be a gamer’s dodecahedron die with 12 sides), we’d not be able to calculate any probability of outcomes. Even though it involves some rudimentary math, the Classical Theory is still inductive. There is not absolute certainty, for instance, that there are only two outcomes to a coin flip. It remains logically possible that it might land on its side, or that it will float in the air and never drop. It could (if we’re talking logical possibility here) turn into an elephant and fly to the Moon. These bizarre, counter-intuitive scenarios are enough to keep probability calculations about ordinary events like die or coin tosses inductive operations. The Classical Theory is easy to use on single, isolated events. We take the number of possible “winners,” or favorable outcomes, and make that the “top” number (i.e., the numerat or) in our probability fraction. We then take the total number of possible outcomes and make that the “bottom” number (i.e., the denominator) of our fraction. And that’s it! That’s the probability of the event taking place, or of “winning”! The formula loo ks like this: P(A) = f/n In English, that says, “The Probability of event A is the number of favorable outcomes over the number of total outcomes.” Below are some illustrations. For the sake of the examples and practice problems in this text, unless it’s stated otherwise, all dice will be six-sided, decks of cards will consist of the 52 normal playing cards (minus jokers) in a poker deck, and all events will be honest (e.g., no loaded or shaved dice). What is the probability of rolling a 2 on one roll of a die? Well, there is only one “winning” or favorable outcome (i.e., 2), so 1 is the top number of
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our fraction: 1/x. Since there are six possible outcomes to our toss of the die, we place 6 at the bottom of our fraction to get 1/6. That’s it! We’re done. What is the probability of rolling an even number on one roll of a die? There are three possible favorable outcomes (2, 4, and 6). So we place a 3 at the top of our fraction to get 3/x. Moreover, there are six possible outcomes, so we place a 6 at the bottom of our fraction to get our answer: 3/6, which reduces to 1/2. What is the probability of selecting a black jack on one honest blind draw from an ordinary deck of 52 playing cards? There are two favorable “winners” (i.e., the jack of spades and the j ack of clubs), so we place a 2 at the top of our fraction. There are a total of 52 cards to draw from, so we place 52 at the bottom of our fraction to get 2/52 or 1/26. Imagine that I toss out the face cards (kings, queens, and jacks) from my deck of cards. What is the probability that you will draw an ace on one blind draw? There are four favorable outcomes (i.e., the four aces), and a total of 40 cards to draw from (there were 12 face cards, leaving 40 numbered cards). So, the probability of drawing an ace from this smaller deck is 4/40, which reduces to 1/10. 203 An opaque urn has three green balls, two black balls, and five yellow balls in it. You reach in on a blind draw and select one ball. What is the probability that it is (a) a black ball? (b) a red ball? (c) any of the balls? Answers: (a) There are two black balls
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and a total of ten balls in the urn, so the probability of drawing a black ball is 2/10 = 1/5. (b) There is no red ball in the urn among a total of ten balls, so the probability of drawing a red ball is 0/10, which equals 0, which means it’s impossible. (c) Here, any ball is a “winner,” so the number of favorable outcomes is ten, and the total number of balls is still ten. So, the probability is 10/10, which is 1, which means you are absolutely guaranteed to win (i.e., to draw a ball). Woo hoo! # The Relative Frequency Theory The Relative Frequency Theory of probability was developed by insurance actuaries in the 18th century as they tried to determine the likelihood of groups of people living to a given age. The Classical Theory could not be used, because the chances of living to age 20, 30, 40, 50, and beyond are not equal. The Relative Frequency Theory is simple, though. The way it works is to observe a number (the larger the better) of outcomes and see how many of those exhibit the particular outcome you are interested in. The handy-dandy formula looks like this: P(A) = f˳/n˳ In English, this says, “The Probability of event A is the number of favorable observed outcomes over the total number of o bserved outcomes.” Don’t get bogged down in what looks like weird math. This is even easier than the Classical Theory. An example will make this fairly clear. Let’s imagine you want to know the probability of 17 -year-old males who have just received their driver’s license getting into an auto accident during their first year of driving. What you do is observe as many 17-year-old males as you can who just received their driver’s
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licenses, and watch them for one year. The more such males you watch for a year, the stronger and more reliable will be your inductive probability calculation. In the course of that year, you note how many of these males get into an auto accident (i.e., the “favorable” outcome; that is, the one you are interested in). If you were able to observe a total of 500 17-year-old males the first year they receive their driver’s licenses, and you observed 25 o f them getting in auto accidents, the probability you are looking for is 25/500 or 1/20. We can also present the results in other terms: 0.05, or 5%, or with odds of 1:19. Here’s another example. You want to know the likelihood of evening shoppers purchas ing a brand of soap now that you —as store manager —have placed it at the end of a store aisle. You watch 1000 shoppers walk past during evening hours, and 25 of them pick up and purchase the soap. The probability is determined simply by making a fraction, placing the favorable observed outcomes number at the top (to get 25/x) and placing the total number of observed outcomes at the bottom, to get 25/1000 = 1/40. And again we can present the probability in other terms: 0.025, or 2.5%, or with odds of 1:39. # The Subjectivist Theory 204 The probability of some events occurring cannot be determined by either the Classical Theory or the Relative Frequency Theory. For instance, what if we wanted to know the probability of the Seattle Seahawks winning their first football game in the upcoming season? Or the probability of Tom marrying Sue this year? The Classical Theory cannot work in either case, because even though there seem to be two clear outcomes (win or lose,
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marry or not), the chances of either happening are surely not equal. Nor can we use the Relative Frequency Theory, because we do not have a background of a total number of observed outcomes; we have not seen the Seahawks play their first game of this upcoming season before (it’s a unique event), and even if Tom and Mary have married in the past, we surely have not observed enough such legal unions between them to use this second theory. Again, this potential marriage is probably a unique event that has not been observed in the past. What’s a gambler to do who wishes to bet on the Seahawks game or Tom and Sue’s relationship? The Subjectivist Theory of probability is the simplest yet, but may seem the least satisfying. What we do is go to an expert, someone who knows the Seahawks better than anyone else, or who has the closest ties to Tom and Sue; we’ll then ask him (we’ll imagine it’s a guy here) what odds he’d honestly, sincerely give that the Seahawks will win or that Tom and Sue will marry. If his odds are sincere and based on his knowledge of the situation, then he should give opposite odds that the Seahawks will lose or that Tom and Sue will not marry. We then take those odds, convert them to a probability, and voilà , we have our probability. Simplicity itself! For formula fans, here is what the odds-probability equivalence looks like: x:y is equivalent to x/(x+y) Let’s have another football example. If we want to do the best we can at determining the probability of the Broncos beating the Raiders in the upcoming game, we go to a football expert who tells us that she’d give 2:1 odds that the Broncos will win (and
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she’d give 1:2 odds that they’d lose). We take those 2:1 odds and translate them into a probability fraction to get 2/3. We now have reason to believe that the Broncos have a 2/3 chance of winning that game! This is called the “Subjectivist Theory” because it is highly subjective to someone’s opinion. No one can know for sure if the Broncos will beat the Raiders, nor can anyone know the exact probability of it happening. There are too many variables involved (sick quarterbacks, psychotic tight ends, a last-minute jailing of the star running back, a drunken coach…and that’s only some bad factors that most people will not know about; good factors can skew probability the other way). Still, some people are more knowledgeable about a team’s chances than are other people, and the odds they honestly and sincerely provide count for more than do those of less informed people. If the experts truly have expertise in the matter, and if they provide us with odds they sincerely believe in, we may not have much else to go on. That is, our understanding of probability in such unique cases may not amount to much more than this. Now let’s have one more example of using the Subjectivist Theory. What’s the probability of Jared getting accepted to Harvard Law School? He’s applied there only this one time, so we have no body of observed outcomes with which to use the Relative Frequency Theory. And the chances of his being accepted or not are surely not dead equal, so we c an’t use the Classical 205 Theory. The best we can do is to find an expert on the matter. Perhaps that is a school councilor who knows Jared and his academic career in high school and college, and who knows something
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about law school admittance demands. If this expert offers an honest appraisal of 1:10 odds, then we can do the simple translation to a probability fraction to determine that from the subjective opinion of the best expert we can find, Jared has a 1/11 chance of getting into Harvard Law School. # **Practice Problems: Single-Event Probability Calculations 1. What is the probability of rolling an odd number other than three on one roll of a six-sided die? What are the odds? 2. What is the probability of drawing an even numbered card on one blind draw of a deck of 52 playing cards? 3. If we watched 200 Bellevue College students take a Symbolic Logic test, and 25 received an A, then on that basis what is the probability of a Bellevue College student receiving an A on a Symbolic Logic test? What are the odds? What is the probability of a Bellevue College student not receiving an A on a Symbolic Logic test? What are the odds? 4. Stan is Bill’s best friend and has known him all Bill’s life. Stan give 3:5 odds that Bill will buy a Ford USV within the year. What probability should we —on this basis —assign to this purchase taking place in this time frame? 5. Take what we know from problem #4 , and include Andy’s odds of 5:1 that Bill will buy a Ford SUV within a year. Andy has just met Bill, and is an enthusiastic, optimistic Ford dealer. What probability should we now assign to Bill’s buying Ford SUV within a year? 6. Imagine an urn with four white balls, three black balls, and one green ball. What is the probability of selecting a white ball on one blind draw, when you can draw one and only one ball
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on any given draw? A black ball on one blind draw? A green ball on one blind draw? What are the odds for each case? What theory needs to be used here? 7. What is the probability of drawing a red ball from the urn in problem #6? What is the probability of drawing a ball on one blind draw from the urn? What are the odds in each case? 8. Janelle gives 10:1 odds that the Cowboys will beat the Broncos in their next game together. Given her odds, what is the probability that the Cowboys will beat the Broncos? What theory needs to be used in this problem? 9. Of 367 Bellevue College students who completed Critical Thinking last year, 192 required psychiatric care within six weeks. Based on those observations, what is the probability that a Bellevue College Critical Thinking student will require psychiatric care within six weeks of completing the course? What are the odds of this happening to such a student? What theory needs to be used in this problem? 10. Imagine an ordinary deck of 52 playing cards. What is the probability of getting a jack or a red queen on one blind draw? What are the odds? 11. Juanita believes that she has a 50% chance of getting an A in her Chemistry class. What probability in terms of a fraction, and then in terms of a decimal number, should she give herself assuming this estimate? What odds should she give herself assuming this estimate? 12. The Bellevue Alimentary Robustness Foundation observed 175 people eating lunch at the Bellevue College Cafeteria. Of those observed, 35 became ill immediately afterwards. BARF has good reason to believe that the food caused the illness. Given this finding, what is the probability 206 that a BC Cafeteria
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diner will get sick due to eating there? What is the probability in terms of a percentage? In terms of a decimal number? What are the odds of getting sick from eating there? 13. Imagine an urn with three green balls, two yellow balls, and ten brown balls. What is the probability of selecting a yellow ball on one blind draw? What would be the odds? What theory needs to be used to determine the probability? 14. Janet believes that the odds are 2:3 that the Eagles will win next week’s football game. Given those odds, what is the probability that the Eagles will lose that game? What theory are you using here? What odds should Janet give that the Eagles will lose ? 15. Julia’s four friends and eight family members (who all took Critical Reasoning) have watched her meet ten football quarterbacks in the past two years, and she has dated seven of them within three weeks. Now that Julia has met the Seattle Seahawk’s first -string quarterback, and given the information provided here, what probability and odds should her friends and family decide for Julia to date this quarterback within three weeks? What theory do they use here? 16. Jessica and her new boyfriend begin to play a rather frisky game of cards, but all of the kings and half of the jacks are missing. At one point, Jessica fans the deck of cards out to her friend, and asks him to select one blindly. He does so. What is the probability that he will select an ace? A jack? A king? Any card at all? What theory are you using here? 17. Janelle has gone to Death Valley 12 times in different summers, and nine times the daily mid-day temperature was over 110 F. She is
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going to Death Valley again this summer. What is the probability that the daily mid-day temperature will not reach 110 F? What are the odds? What theory are you using to determine the probability? 18. Jackie places a knight randomly on an empty chess board. What is the probability that she places it on a square on which a pawn usually sits at the beginning of a game of chess? What is the probability in terms of a decimal number? In terms of a percentage? What are the odds? What theory are you using to determine the probability? 19. Imagine that Jackie is playing white and is about to make her first move. She knows nothing about chess theory, so her move is legal but random. What is the probability that her first move will be to place one of her men on c3 (i.e., the square directly in front of her queen’s bishop)? What are the odds? 20. Jillian had three shots of bourbon while playing craps at a Las Vegas casino on Tuesday and won money at the game. She had three shots of Scotch while playing roulette at the casino on Wednesday and won money at the game. On Thursday she drank three shots of rum and lost money to the casino at blackjack. On Friday she drank three shots of gin and won money at baccarat. It’s now the following Monday, and she’s about to play poker. Jillian quickly drinks three shots of tequila, using what she takes to be critical thinking skills and the Relative Frequency Theory to conclude that her probability of winning is greater with three shots of alcohol in her. What probability did she derive for winning money at poker while mildly besotted this evening? Why is her reasoning and use of
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the Relative Frequency Theory poor? 21. A dishonest gambler shaves a die slightly on one particular side so that it’s more rectangle than square, making the die more likely to come up on one face rather than another. What theory would you use to determine the probability of the die coming up an even number? 22. Washington Senator Sunny Shine is entering her first re-election campaign. Sh e thinks she’s got a pretty good chance of retaining her Senate seat. Pastor Bustle and some of his parishioners are vehemently against Shine’s public support of a clothing -optional beach at a remote 207 Washington State Park, so they form the Public Opposed to Reckless Nudity political action committee to oppose Shine’s re -election. Bustle has stated in media interviews that “Shine has no chance at all of re-election!” How shall we best determine the probability of Shine getting re -elected? 23. You are playing some form of draw poker and see three aces, four kings, two queens (all face up), and ten other unknown cards face down on the table. Your three friends hold five cards each, none of which your friends let you see. You need a queen to fill out a straight (K-Q-J-10-9) you’re seeking, so you discard one useless card and draw one card from the deck. What is the probability of getting your needed queen? Answers: 1. 2.6 = 1/3, 1:2 2. 20/52 = 5/13, 5:8 3. 25/200 = 1/8, 1:7, 175/200 = 7/8, 7:1 4. 3/8 5. It’s st ill 3/8, because Stan is more of an expert on Bill than is Andy. 6. 1/2, 3/8, 1/8, 1:1, 3:5, 1:7, Classical Theory 7. 0/8, 1/1, 0:8, 1:0 8. 10/11, Subjectivist Theory 9. 192/367, 192:175, Relative Frequency Theory 10. 3/26, 3:23 11. 1/2, 0.5, 1:1 12. 1/5,
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20%, 0.2, 1:4 13. 2/15, 2:13, Classical Theory 14. 3/5, Subjectivist Theory, 3:2 15. 7/10, 7:3, Relative Frequency Theory 16. 2/23, 1/23, 0, 1, Classical Theory 17. 1/4, 1:3, Relative Frequency Theory 18. 1/4, 0.25, 25%, 1:3, Classical Theory (there are 16 pawns at the beginning of a game, and a chess board has 64 squares of equal size) 19. 1/32, 1:31 (she can move her queen’s knight or her queen’s bishop pawn there) 20. 3/4. There are many problems with Jillian’s thinking, but two major concerns are her embracing a False Cause fallacy (drinking alcohol does not cause people to win at games of chance), and the uncertainty that she has equal knowledge, skill, or experience with each game. 21. Since we do not know the exact amount shaved off the die, and thus cannot determine the likelihood of each face coming up, we cannot use the Classical Theory. We can, however, roll the die 100 times and see how many times it comes up even. We’d thus be using the Relative Frequency Theory to calculate a fairly strong probability. If we rolled the die 1000 times and counted the number of times it came up even, we’d have an even stronger argument for the probability. 22. There are two outcomes (re-elected or not re-elected), but the chances are not equal. So we can’t use the Classical Theory. Shine has never entered a re-election campaign before, so we have no set of observed experiences with some favorable outcomes to draw upon. Thus we cannot use the Relative Frequency Theory. We are left with the Subjectivist Theory, but Pastor Bustle and PORN are likely unreliable sources of odds, especially as Bustle speaks emotively before the media. Shine’s opinion of her chances may be somewhat biased or overly optimistic, 208 so
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she may not be a reliable source of odds, either. What we’d n eed to do is find an expert: someone who knows Shine well, the mood of Washington voters, and the circumstances underlying the re-election. We’d seek odds of Shine winning from this expert, and calculate Shine’s probability of winning based on that. That m ay be the best we can do here. 23. We can use the Classical Theory here, but it’s a bit complicated getting there. There are 14 cards you know the nature of: three aces, four kings, two queens, and the five in your hand (none of which is a queen). The ten unknown cards on the table, the five your friends are each holding close to their chests, and the remaining cards in the deck make up the pool from which you hope to get a queen (of course you can draw only from the deck). The deck itself thus now contains 13 cards (52-14-10-5-5-5=13). There are two queens left somewhere among the unknown cards, so you have a probability of 2/13 of getting a queen. This assumes that none of your friends is neurotic about keeping or discarding queens, and that you have no good reason to think any friend is holding on to one to better his or her hand. Since queens are generally more valuable than most other cards, we might want to say that the Classical Theory can tell us that you have no better than 2/13 chance of drawing a queen here. Gambling is clearly fraught with peril, and students would do best to avoid it at all costs. They will probably be safer watching television. Wanna bet on that? # Probability Calculations Each probability problem we’ve looked at so far concerns the probability of a single result from
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a single event: one number coming up on a roll of a die, one draw from a deck of cards producing specified card, male drivers getting into an auto accident, Bob and Sue getting married, or the Seattle Seahawks winning their next game. Sometimes, however, we want to determine the probability of more than one event happening, or the probability of either of two events happening. For instance, we may want to know the probability of drawing an ace and then a king from a deck of cards, or we may want to know the chances of getting two sixes on a roll of a pair of dice. Or, we may want to know the likelihood of Sue marrying Bob or Tim, or the probability of getting a king or a queen on a draw from a deck of cards. These slightly more complex calculations require one or more additional rules. We’ll examine a small handful here to help us with some fairly straightforward calculations. A wee bit of grade-school math is needed, but no more than to add or multiply some basic fract ions. It’s do -able. Feel free to keep a calculator handy for the practice problems, though. # Restricted Conjunction Rule Let’s imagine that we need heads on two tosses of a coin. We toss the coin once and get heads. Good so far! We toss it a second time and get heads again. We win! What was the probability of getting heads those two times in a row? To determine this, we make use of the Restricted Conjunction Rule (RCR). It’s a “conjunction” rule because it’s a rule about the conjunction of two events: we want to know the probability of getting the first heads and getting the second heads. To use the rule we simply
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determine the probability of getting heads the first time and multiply that by the probability of getting heads the second time: 1/2 x 1/2 = 1/4. This result should sound correct, as there were four possible outcomes for the two coin tosses: 209 H-H H-T T-H T-T There was thus a one-in-four chance of getting two heads; that is, there was a probability of 1/4. A formula for the RCR looks like this: P(A and B) = P(A) x P(B) This says —in fairly normal English —that the probability of events A and B both occurring equals the probability of A occurring times the probability of B occurring. We can use the Classical, Relative Frequency, or Subjectivist Theories to determine each individual probability (depending on which theory is called for), and then simply multiply them. RCR works only when two or more events are independent of each other. Events are independent of each other when they do not impact or affect each other. Two coin tosses have no effect on the outcome of each other, nor do two separate rolls of a single die. Imagine, however, wishing to draw two aces from a deck of 52 playing cards when you do not replace the first card in the deck prior to making the second draw. The first draw would have an impact on the probability of getting that second ace because the deck now would have only three aces and a total of 51 cards. We’ll need a second conjunction rule for this scenario. But first, let’s lo ok at some more examples of finding the probability of two events that are independent .What is the probably of rolling a six-sided die twice and getting a five both times? We roll the die the first time with a 1/6 probability
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of getting a five. Let’s sa y we get that five! We roll the die again and get the second five. The probability of getting that five was again 1/6. So, we multiply the two probabilities to get our final result: 1/6 x 1/6 = 1/36. What’s the probability of getting three heads in a row on three tosses of a coin? Each independent toss resulting in heads has a probability of 1/2. So we multiply each of the three probabilities to get our final result: 1/2 x 1/2 x 1/2 = 1/8. What is the probability of rolling one six-sided die four times, and getting a two each time? Well, the probability of getting a two on any one of those rolls is 1/6, so to find the probability of getting four winners in a row we multiple 1/6 four times: 1/6 x 1/6 x 1/6 x 1/6 = 1/1296. Finally, since urns with balls in them are so cool, let’s imagine that we have an opaque urn holding three red balls, two green balls, and five black balls. We want to know the probability of drawing two black balls, when we replace the first ball drawn before reaching in for the second ball. The draws are independent because we replace that first drawn ball before drawing the second. If we did not replace the first ball, the first draw would have an impact on the probability of the second draw (because there would be four black balls in the urn instead of the original 210 five, and there would be a total of nine balls in the urn instead of the original ten). So, we reach in and draw the first ball. The probability of getting a black ball is 5/10, or 1/2. We replace that ball, shake
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the urn a bit, reach back in and fortuitously draw a second black ball. The probability of drawing that ball is also 1/2. To make our final determination of the conjunction of drawing the first black ball and the second black ball, we multiply the two probabilities: 1/2 x 1/2 = 1/4. # **Practice Problems: Restricted Conjunction Rule Determine the probability of the following set of events using the Restricted Conjunction Rule. 1. What is the probability of getting five heads in a row on five tosses of a coin? What are the odds of this happening? 2. What is the probability of blindly drawing two black jacks from a deck of 52 playing cards when you replace the first card drawn before making the second draw? What are the odds? 3. Imagine an urn containing one white ball, two yellow balls, and one blue ball. What is the probability of drawing a white ball twice on two independent blind selections (i.e., when you replace the first ball drawn before drawing the second ball)? What are the odds? 4. Close friends of Marko and Valeria give the couple 1:2 odds of getting married within the month. What is the probability that the couple will get married within the month and you get tails on an honest toss of a coin? What are the odds? 5. What is the probability of rolling two sixes (“boxcars”) with one toss of a pair of six -sided dice? What are the odds? 6. What is the probability of getting a seven and then an even number on two rolls of a six-sided die? 7. What is the probability of getting a number each time on two rolls of a six-sided die? 8. What is the probability of rolling a three on one roll of
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a six-sided die and drawing an ace from a deck of 52 cards? Answers: 1. 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/32; 1:31 2. 2/52 x 2/52 = 1/676; 1:675 3. 1/4 x 1/4 = 1/16; 1:15 4. 1/3 x 1/2 = 1/6; 1:5 5. 1/6 x 1/6 = 1/36; 1:35 6. 0/6x 3/6 = 0/6 = 0 (it’s impossible) 7. 6/6 x 6/6 = 1/1 = 1 (it’s guaranteed) 8. 1/6 x 4/52 = 1/6 x 1/13 = 1/78 # General Conjunction Rule We need to handle the probability of two events a little differently when the probability of one is dependent on that of the other. For instance, imagine that you need to draw two kings from a deck of 52 playing cards. You draw one and place it in your pocket. The probability of getting that first king was 4/52, or 1/13. You get ready to draw a second time, but now the deck has only three kings in it and the deck itself contains only 51 cards. The probability of getting a king now is 3/51. The two probabilities together (i.e., conjoined) equals 1/13 x 3/51 = 1/221. If we 211 assumed inco rrectly that the probability each time was the same (i.e., 1/13), then we’d get 1/13 x 1/13 = 1/169. That’s a different (mistaken) answer altogether. The General Conjunction Rule (GCR) is used when we want to know the probability of two or more events when the earlier event has an impact on later events. A formula for this rule looks like this: P(A and B) = P(A) x P(B given A) This says that the probability of both A and B occurring equals the probability of A occurring times the probability of B occurri ng (given that A already
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occurred). Let’s look at some examples. Imagine an urn holding two white balls, eight green balls, and two purple balls. What is the probability of drawing two green balls when you don’t replace the first ball drawn? Note that the second draw is dependent on the first, because on the second draw there is one less ball in the urn than on the first draw. The probability of the first draw is determined using the Classical Theory: there are eight possible “winners” and 12 balls total f or a probability of 8/12, or 2/3. For the second draw, we again use the Classical Theory, noting that there are now seven possible “winners” and only 11 balls total. The probability for this second draw by itself is thus 7/11. To determine the probability of getting both green balls, we multiply the two probabilities: 2/3 x 7/11 = 14/33. If we had put the first green ball drawn back in the urn before making our second draw, then the first draw would have had no impact on the second draw. The second draw would thus be independent of the first, and we could use the Restricted Conjunction Rule: 2/3 x 2/3 = 4/9. GCR is needed when the probability of one event is dependent on another; RCR can be used when the outcomes of two (or more) events are independent of each other. Note, though, that GCR may be used in any conjunction calculation (RCR may be easier to use, however, when the events are independent). Let’s go back to the urn problem immediately above, and once again replace the first green ball drawn before making our second draw. We can use GCR here instead of RCR. The probability of drawing the first green ball is 2/3. We’ll now replace it. Given
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the first probability, and given that we just replaced the ball, the probability of drawing a second green ball is also 2/3. So we get 2/3 x 2/3 = 4/9. So, students could —if they wished to know the minimum needed to get by with calculating the probability of conjoined events —learn only the General Conjunction Rule. It’s called a general rule because it may be used to determine the probability of the conjunction of both dependent and independent probabilities. We need more examples! What is the probability of drawing two black queens from a normal deck of cards, assuming no replacement of the first card drawn? Wel l, the first probability is 2/52, or 1/26. That’s because there are two black queens in the deck of 52 cards. For the second draw, there is only one black queen left, and only 51 cards in the deck. The probability of that second draw is thus 1/51. We 212 then multiply the two probabilities to calculate the probability of the two conjoined events: 1/26 x 1/51 = 2/1326 = 1/663. Imagine a deck with nothing but face cards (kings, queens, and jacks) of all four suits (clubs, hearts, spades, and diamonds). What is the probability of drawing three jacks when you do not replace each card drawn? The probability of the first draw is 4/12, or 1/3. We now place the jack over to the side, and draw another card. The probability of getting a second jack is 3/11. We set that jack aside, and draw a third time. The probability of getting a third jack (given that we’ve drawn two previously) is 2/10, or 1/5. We now multiply the three probabilities: 1/3 x 3/11 x 1/5 = 3/165 = 1/55. Imagine an urn containing two brown balls, one pink ball,
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and seven orange balls. What is the probability of drawing a brown ball (without replacing it afterwards) and then a pink ball? To figure this out, we begin by determining the probability of the first draw. There are two “winning” brown balls amid st a total of ten balls. The probability of drawing a brown ball at this point is thus 2/10, or 1/5. We now set that brown ball aside and reach in for our second draw. We now have nine balls in the urn with one pink “winner” possible. The probability of dr awing the pink ball at this point is 1/9. We then multiply the two probabilities and get our final result: 1/5 x 1/9 = 1/45. # **Practice Problems: General Conjunction Rule Determine the probability of the following events using the General Conjunction Rule. 1. Imagine an urn with three red balls, two blue balls, and one green ball. What is the probability of drawing two red balls on two blind draws, without replacement? 2. Imagine the same urn as in problem #1 above. What is the probability of drawing three red balls on three blind draws, without replacement? 3. Imagine the same urn as in problem #1 above. What is the probability of drawing a green ball and then a red ball, without replacement? 4. Imagine the same urn as in problem #1 above. What is the probability of drawing a red and then a purple ball on two blind draws, without replacement? 5. Once again, imagine the same urn as in problem #1 above. What is the probability of drawing a ball on each of two blind draws, without replacement? 6. Imagine an ordinary deck of 52 playing cards. What is the probability of drawing a black king and then a red card on
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two blind draws, without replacement? 7. Three logic professors are relaxing at hotel bar after a long day attending a philosophy conference on symbolic logic. They drink too much, and stumble their way one after the other attempting to go back to their individually assigned rooms. There are only three rooms (each presently unlocked) in this hotel, and there is an even chance of each man lurching his way to any one of the rooms. (The female logicians at the conference are whooping it up elsewhere having what they’re calling “Ladies’ Night.”) Once each inebriated man gets to a hotel room, he locks the door behind him and falls unto his bed in a stupor. The men leave the bar one by one, each stumbling around looking for an open room. What is the probability that each will end up in his assigned room? 213 Answers: 1. 3/6 x 2/5 = 2/10 = 1/5 2. 3/6 x 2/5 x 1/4 = 2/40 = 1/20 3. 1/6 x 3/5 = 3/30 = 1/10 4. 3/6 x 0/5 = 0/30 = 0 (there are no purple balls in the urn, so such a sequence of draws is impossible) 5. 6/6 x 5/5 = 30/30 = 1 (it’s guaranteed that you’ll draw some ball of one color or another on each draw) 6. 2/52 x 26/51 = 52/2652 = 13/663 7. The probability of the first guy making it to his room is 1/3. Let’s assume he gets there. The second guy now has a probability of 1/2, as there is one “winning” room and he has two options left (the first guy locked his door and is now passed out on his assigned bed). Given that all this takes place, the third guy has only one room left: his room. So the
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probability of his getting into that one is 1/1, or 1. We multiply all three probabilities together for the final result: 1/3 x 1/2 x 1/1 = 1/6. # Restricted Disjunction Rule If we want to know the probability of two events both taking place (“conjoined” as it were), we’d use a conjunction rule. But sometimes we want to know the probability of event A happening or event B happening. Here we need a disjunction rule. There are two such rules we can use, with the first being a little easier, but applying to a rather restricted set of circumstances. The Restricted Disjunction Rule (RDR) is used when we want to know the probability of one or another event occurring, and when the events are mutually exclusive . Two events are mutually exclusive if and only if they cannot both take place (although it might end up that neither takes place). Either occurs, but not both. Examples of mutually exclusive events include rolling one die and getting either a two or a three, drawing either a king or queen on one blind draw from a deck of cards, and getting heads or tails on a toss of a coin. In each case, you can get one result, but not both (and sometimes neither). Whereas for the probability of the conjunction of two events we multiply the probability of each event, for disjunction calculation we add the two probabilities. The formula for RDR looks like this: P(A or B) = P(A) + P(B) This says that the probability of either A or B occurring is equal to the probability of A occurring plus the probability of B occurring. As always, examples will help. What is the probability of rolling an even number or a three on one honest roll of a
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six-sided die? To determine this, we first figure out the probability of getting an even number. There are three “winners” (i.e., two, four, and six), so the probability is 3/6. The probability of getting a three is 1/6. So, the probability of getting an even number or a three is 3/6 + 1/6 = 4/6 = 2/3. 214 What is the probability of drawing a red king or a spade on one blind draw from a deck of 52 playing cards? It doesn’t really matter which probability we determine first, so let’s just start with the red king. There are two “winners” (i.e., two red kings: the king o f diamonds and the king of hearts), so the probability of drawing one is 2/52. There are 13 spades in the deck, so the probability of drawing one of them is 13/52. The probability of drawing one or the other (and we can’t get both with one draw, so they ar e mutually exclusive) is 2/52 + 13/52 = 15/52. Imagine we have an urn with two red balls, one blue ball, four yellow balls, and three green balls. On a blind draw we select one ball. What is the probability that ball will be green or red? The probability of selecting a green ball is 3/10, and the probability of selecting a red ball is 2/10. So adding those two probabilities together will give us the probability of selecting one or the other: 3/10 + 2/10 = 5/10 = 1/2. That should be intuitive, as the red and green balls together make up half the number of balls in the urn, so we should have a 1/2 chance of getting one or the other. Imagine the same urn as above. What is the probability of selecting on one blind draw
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a red, blue, or yellow ball? Here we have three mutually exclusive events, as we can draw only one ball. The probability of getting a red ball is 2/10; the probability of getting a blue ball is 1/10; the probability of getting a yellow ball is 4/10. Adding them up, we get the probability of getting one of them: 2/10 + 1/10 + 4/10 = 7/10. # **Practice Problems: Restricted Disjunction Rule Determine the probability of the following mutually exclusive events. Use the Restricted Disjunction Rule. 1. What is the probability of drawing an ace or an even numbered card with one blind draw from a normal deck of 52 playing cards? 2. What is the probability of selecting on one blind draw a face card, a ten or, a red two from a deck of 52 playing cards? 3. What is the probability of rolling an even or odd number with a six-sided die? 4. Imagine an urn containing two red balls, four black balls, and three white balls. What is the probability of drawing either a white or black ball on one blind draw? 5. Imagine an urn with one yellow ball, two blue balls, two green balls, three red balls, and two purple balls. What is the probability on one blind draw of selecting either a yellow, blue, or green ball? 6. We watched ten prisoners at County Jail eat lunch, and three freely chose to eat cheeseburgers (which are served every day). We also watched 20 other County Jail inmates eat lunch, and of that group three freely chose to eat tuna sandwiches (which are served every day). Inmates at County Jail must eat lunch, but may choose only one item for lunch. Given this information, what is the probability that a County Jail inmate will
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eat either a cheeseburger or tuna sandwich for lunch? 7. Janet is practicing poker by herself in her hotel room. She takes a deck of 52 playing cards and draws four cards: the ace of hearts, the two of diamonds, the three of clubs, and the four of hearts. She’s about to draw one more card, but pauses to determine the probability of drawing another ace to get a pair of aces, or any five to get a straight. What is that probability? 215 Answers: 1. 4/52 + 20/52 = 24/52 = 6/13 2. 12/52 + 4/52 + 2/52 = 18/52 = 9/26 3. 3/6 + 3/6 = 6/6 = 1 (it’s guaranteed) 4. 3/9 + 4/9 = 7/9 5. 1/10 + 2/10 + 2/10 = 5/10 = 1/2 6. Using the Relative Frequency Theory, we determine the probabilities of County Jail inmates eating cheeseburgers or tuna sandwiches for lunch: 3/10 and 3/20 respectively. We then use RDR to get the probability of inmates eating either item for lunch: 3/10 + 3/20 = 6/20 + 3/20 = 9/20. 7. The probability of getting a second ace is 3/48 (since there are three aces left in the deck that now has 48 cards in it). The probability of getting any five is 4/48 (since there are four fives in the remaining deck of 48 cards). On one draw Janet can’t get an ace and a five (they are mutually exclusive), so she uses RDR to get the probability of drawing either of the desired cards: 3/48 + 4/48 = 7/48. # General Disjunction Rule The Restricted Disjunction Rule works for situations in which two potential events are mutually exclusive, like rolling a two or a three with one roll of a die. The events are mutually exclusive because you can’t get
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both numbers on one roll. But some pairs of potential events are not mutually exclusive; that is, one or the other, or both might occur. For instance, let’s say you are about to d raw a card from a full deck of playing cards and will “win” if you get either an ace or a spade. You could win with an ace, or a spade that’s not an ace, or the ace of spades. Drawing an ace does not exclude you from drawing a spade; you might get both. To calculate the probability of drawing an ace or a spade (or both), we use the General Disjunction Rule (GDR). GDR calculates the probable occurrence of either of two or more independent events whether or not they are mutually exclusive. (If you highlight textbooks, you might want to highlight that last sentence.) Since RDR is easier to work with, it’s usually best to simply use it when the two events are mutually exclusive. We need to use GDR when the two events are not mutually exclusive. To understand the idea behind GDR, continue to imagine that you hope to draw an ace or spade (as above) from a full deck of cards. Either card will be a winner (and of course so will the ace of spades). We might be tempted to add up the two possibilities of getting an ace or a spade to get a final calculation. There are four aces, so the probability of getting an ace is 4/52. There are 13 spades, so the probability of getting a spade is 13/52. But merely to add these two probabilities (getting 17/52) would be a mistake, be cause one of the spades is an ace, and we’d be counting it twice. So, we need to account for this “overlap”
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caused by aces and spades not being exclusive of each other, and subtract the ace-of-spades probability (i.e., the probability of drawing an ace and a spade on one draw: 13/52 x 4/52 = 1/52) from the sum of the ace and spade probabilities. Our calculation will then look like this: 13/52 + 4/52 – 1/52 = 16/52 = 4/13. This example illustrates the GDR formula: 216 P(A or B) = P(A) + P(B) – P(A x B) This formula says that the probability that either of two independent events —mutually exclusive or not —equals the probability of one plus the probability of the other minus the probability they both occur together. Let’s look at another simple example. What is the probability of getting at least one tails on two tosses of a coin? We “win” if we get tails on the first toss, the second toss, or on both tosses. The tosses and their results are independent, as the first toss has no impact on the second. Also, the result of tails is not exclusive, as we might get tails on either toss, or on both . Looking at the formula, let’s consider the first toss as event A, and the second toss as event B. The probability of getting at least one tails will thus be calculated like this: 1/2 + 1/2 – (1/2 x 1/2) = 1 – 1/4 = 3/4. And that result should seem fairly intuitive, as there are three ways to win amid the four possible coin-toss scenarios: H-H loses H-T wins! T-H wins! T-T wins! For another example, consider the logician’ s favorite piece of pottery, the urn. This urn contains three red balls, three yellow balls, two green balls, and two black balls. What is the probability of selecting a red ball on
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a blind draw when you get two draws and you have to replace the first ball drawn before drawing the second time? (These latter conditions keep the events independent.) We can win here by drawing a red ball on the first try, on the second try, or on both tries. Selecting red balls each time is thus not mutually exclusive. The probability of a getting a red ball on the first draw is 3/10, and since we replace the ball before drawing again, the probability of getting a red ball on the second draw is also 3/10. Since these events are independent and not mutually exclusive, we use GDR: 3/10 + 3/10 – (3/10 x 3/10) = 6/10 – 6/100 = 60/100 – 6/100 = 54/100 = 27/50. The crowd wants one more example! What is the probability of rolling a pair of dice and getting at least one one? One die could come up one; the other could come up one, or both might come up one giving you “snake eyes.” The events are independent, as the roll of one die does not impact the result of the other roll. And since the disjunction is inclusive (i.e., not exclusive), we’ll use GDR to calculate the probability here. T he probability of getting a one for one toss of one die is 1/6. So, GDR tells us to calculate things this way: 1/6 + 1/6 – (1/6 x 1/6) = 2/6 – 1/36 = 12/36 – 1/36 = 11/36. GDR works whether the events are mutually exclusive or not, but RDR is easier to use when the events are exclusive. Consider rolling a die once again, this time hoping to get a one or a two. Since we’re rolling the die only once, we can’t get both numbers, so the
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results are exclusive: it is impossible for both to occur. The GDR formula can work here, but it’s easier to use RDR. 217 Using GDR, we get the following calculation (keep in mind that getting a one and a two on one roll of a die is impossible, and thus has a probability of zero): 1/6 + 1/6 – (one and two) = 2/6 – 0= 2/6 = 1/3. Sinc e the events are mutually exclusive, we’d be warranted in using the simpler RDR to make the following briefer calculation: 1/6 + 1/6 = 2/6 = 1/3. # **Practice Problems: General Disjunction Rule Determine the probability of the following independent events, noting that they are not mutually exclusive. Use the General Disjunction Rule. 1. What is the probability of tossing a coin twice and getting at least one heads? 2. What is the probability of drawing at least one jack on either of two draws from a normal deck of 52 playing cards, assuming you replace the first card before drawing the second? 3. You roll a pair of dice. What is the probability that at least one will turn up with an even number? 4. Imagine an urn with two green balls, three brown balls, and five orange balls. You draw twice, replacing the first ball before making the second draw. What is the probability of getting at least one green ball? 5. Given the urn in problem #4 above, what is the probability of drawing at least one brown ball, given two draws and replacing the first ball before making the second draw? 6. Assuming that the odds of the Seahawks winning their next football game are 3:2, and that the odds of the Sounders winning their next soccer game are 1:2, what is the probability that either
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will happen? 7. Five women attending a philosophy conference on symbolic logic are spending the later part of an evening at a women’s dance club. Three scantily clad guys are dancing on the stage, and Sarah, one of the five women, shouts, “I know th ose guys! One is a blithering idiot, but is really fun to hang out with; the other two studied logic in college.” She goes on to say that the two brighter lads would surely want to talk about symbolic logic with the women after their dance routine. She says the odds are 3:1 that Thomas will want to do so, and 2:1 that Fabio will want to. Of the remaining four women, only Barbara is sober enough to calculate the probability that either Thomas or Fabio (or both) will want to join them after their dance routine to talk about symbolic logic. What is that probability? 8. Janet and her four philosopher friends eventually leave the dance club and go to a nearby bar to play poker. Janet tries to explain some of the rules to them, and illustrates what she says by drawing four cards randomly from a shuffled deck. She deals herself two kings, an ace, a four, and a six. She discards the four and six and keeps her two kings and the ace. Taking this as an opportunity to show how well logicians can do probability calculations, she asks the four women what the probability is of her getting at least one more king upon drawing two cards (without replacement). The exhausted but contented women all ably come to the same conclusion. What is it? Answers: 1. 1/2 + 1/2 – (1/2 x 1/2) = 1 – 1/4 = 3/4 2. 4/52 + 4/52 – (4/52 x 4/52) = 1/13 +
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1/13 – (1/13 x 1/13) = 26/169 – 1/169 = 25/169 3. 1/2 + 1/2 – (1/2 x 1/2) = 1 – 1/4 = 3/4 4. 2/10 + 2/10 – (2/10 x 2/10) = 1/5 + 1/5 – (1/5 x 1/5) = 2/5 – 1/25 = 10/25 – 1/25 = 9/25 218 5. 3/10 + 3/10 – (3/10 x 3/10) = 6/10 – 9/100 = 60/100 – 9/100 = 51/100 6. We first convert odds to probability: 3:2 is 3/5, and 1:2 is 1/3. Given the Seahawks and the Sounders could both win, we use GDR for our calculation: 3/5 + 1/3 – (3/5 x 1/3) = 9/15 + 5/15 – 3/15 = 11/15. 7. We begin by using the Subjectivist Theory to determine the probability of Thomas and Fabio each joining the ladies in critical dialectic. The odds for Thomas’s future presence are 3:1, so the probability of his joining the ladies is 3/4. The odds for Fabio conferring with the philosophers on matters pertaining to symbolic logic are 2:1, which makes the analogous probability 2/3. Since either or both might wish to chat with the women, we use GDR: 3/4 + 2/3 – (3/4 x 2/3) = 9/12 + 8/12 – 6/12 = 17/12 – 6/12 = 11/12. The women are delighted at Barbara’s findings and expect the evening will progress marvelously. 8. There are 47 cards left in the deck, and within it there are a total of two “winning” kings. So, the probability of getting a “winner” on eit her or both draws (and she could get a winner on both draws, making the results independent but not mutually exclusive) is calculated with GDR: 2/47 + 2/47 – (2/47 x 2/47) = 188/2209 – 4/2209 = 184/2209. # Negation Rule The Negation Rule is easy to
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understand, and is often useful when the conjunction or disjunction rules either won’t work or would be difficult to apply. The idea behind this rule is to figure out the probability of an event not occurring, and then subtract that probability from 1 to get the probability of the event occurring. Assuming that the event will either occur or not occur, the probability of each adds up to 1 (remember that a probability of 1 means an event is guaranteed to occur, just like a probability of 0 means that the event is impossible). The formula for the Negation Rule is P(A) = 1 – P(not-A) This says that the probability of an event A is 1 minus the probability that A does not occur. A fairly simple illustration of the use of the Negation Rule appeals to coin tosses. What would be the probability of getting at least one heads on three tosses of a coin? We could “win” by getting heads on the first toss, the second, the third, the first and second, the first and third, the second and third, or on all three tosses. Whew! That would be a somewhat complex set of disjunctions to calculate. But, there is only one way to “lose,” and that’s by getting tails three times independently in a row (i.e., tails and tails and tails). We can use RCR to determine that easily: 1/2 x 1/2 x 1/2 = 1/8. So the probability of “losing” is 1/8. Using the Negation Rule, we can now easily determine the probability of “winning”: 1 - 1/8 = 7/8. For another example, imagine an urn containing two red balls, two blue balls, two green balls, and four white balls. What is the probability of drawing at least one red, blue, or green ball on
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two tries, when you replace the first ball before drawing the second time? Well, we could use GDR as the events are independent and not mutually exclusive (you might draw a “winning” ball either or both times), but it would be a fairly complicated calculation. We can instead easily determine the probability of “losing,” that is, of drawing a white ball on each draw (which is the 219 only way we can “lose” here). To calculate gettin g a white ball and then another white ball, we use RCR: 4/10 x 4/10 = 2/5 x 2/5 = 4/25. Now we use the Negation Rule to get the probability of “winning”: 1 – 4/25 = 25/25 – 4/25 = 21/25. The Negation Rule also makes it easier to calculate the probability of either of two events occurring that are dependent . Consider the same urn described immediately above. What would be the probability of drawing at least one red, blue or green ball on two blind draws when you do not replace the first ball drawn before se lecting the second ball? Again, the only way to “lose” here is to draw a white ball and then another white ball. We can use GCR this time to calculate with relative ease this probability of “losing.” Note that after drawing a “losing” white ball on the first draw that there are only three white balls left among a total of nine balls. GRC gives us the following calculation: 4/10 x 3/9 = 2/5 x 1/3 = 2/15. So, we stand a 2/15 chance of “losing.” We now use the Negation Rule to determine the probability of “winning”: 1 – 2/15 = 15/15 – 2/15 = 13/15. Cooly cool! # **Practice Problems: Negation Rule Using the Negation Rule as part of your calculation,
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determine the probability of the following events. 1. What is the probability of getting at least one tails on four tosses of a coin? 2. What is the probability of getting at least one two on three rolls of a six-sided die? 3. Imagine an urn with one black ball, five yellow balls, and two green balls. What is the probability of drawing at least one black or yellow ball given two draws when you do not replace the first ball drawn before making the second draw? 4. Consider the same urn as in problem #3 above. What is the probability of drawing at least one black or yellow ball given two draws when you do replace the first ball drawn before making the second draw? 5. (a) What is the probability of drawing at least one king from a deck of 52 playing cards, if you are given two chances to do so, and you must replace the first card drawn before drawing the second? (b) What is the probability if you do not replace that first card before drawing the second? 6. Only three racehorses —Bellevue Slew, Administariat, Woman O’ War— are competing at Emerald Downs in two races. The odds of winning for each horse in the first race are Bellevue Slew 1:2, Administariat 1:5, and Woman O’ War 1:1. For the second race, the odds are, respectively, 3:2, 1:4, and 1:4. Given these odds, what is the probability that either Bellevue Slew or Administariat will win at least one race? What are the odds of this happening? Assume that the first race and its results have no impact on the second race and its results. 7. Three male and five female philosophy professors wake up with hangovers while attending a conference on symbolic logic. Blurry-eyed and
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aching, each stumbles haltingly to the hotel café for coffee. Once there, the waitress says that the café has a new pricing option for their $1 cups of coffee. Each customer can roll three six-sided dice, and if at least one die shows a six, then the coffee is free; if not, then the customer pays $2 for a cup of coffee. The men all jump at the chance, figuring it’s a fun, even bet and, frankly, they’re in too much pain to think about it. Among the women, Janet is more cautious and wants to calculate the probability of getting a six 220 with a little more precision than generated by some hung-over guy’s blurred intuition. What is the probability of getting at least one six on a roll of three dice? Is this an even bet? Answers: 1. P(H and H and H and H) = 1/2 x 1/2 x 1/2 x 1/2 = 1/16; P(at least one T) = 1 – 1/16 = 15/16. 2. P(not-2 and not-2 and not-2) = 5/6 x 5/6 x 5/6 = 125/216; P(at least one 2) = 1 – 125/216 = 91/216. 3. The only way to “lose” is to draw two green balls. We find the probability of that by using GCR (since the second draw is not independent): 2/8 x 1/7 = 2/56 = 1/28. We now use the Negation Rule to determine the probability of “winning”: 1 – 1/28 = 27/28. 4. Again, the only way to “lose” is to draw two green balls. We find the probability of that b yusing RCR (since the second draw is independent): 2/8 x 2/8 = 4/64 = 1/16. We now use the Negation Rule to determine the probability of “winning”: 1 – 1/16 = 15/16. 5. (a) There are 48
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cards in the deck that are not kings; the only way to “lose” here is to draw a non-king twice. For the first scenario, we replace the first card drawn before making the second draw. Since this second draw is independent of the first, we can use RCR to determine the probability of drawing a non-king: 48/52 x 48/52 = 12/13 x 12/13 = 144/169. We now use the Negation Rule to determine the probability of “losing” (i.e., of getting something other than a non-king, i.e., a king): 1 – 144/169 = 25/169. (b) For the second scenario, we do not replace the first card drawn before drawing the second, so here we use GCR to determine the probability of “losing” (i.e., drawing two non -kings): 48/52 x 47/51 = 2256/2652 = 188/221. Now we use the Negation Rule to calculate the probability of “winning” (i.e., getting a king at least once): 1 – 188/221 = 33/221. 6. First we use the Subjectivist Theory, and translate the odds into probabilities. Each horse has two races and two probabilities: the probabilities of Bellevue Slew winning are 1/3 and 3/5; Administariat’s are 1/6 and 1/5; Woman O’ War’s are 1/2 and 1/5. The only way we can “lose” is for Woman O’ War to win both races. Since the two races are independent, we can use RCR to determine the probability of “losing”: 1/2 x 1/5 = 1/10. Now we use the Negation Rule to determine the probability of “winning”: 1 – 1/10 = 9/10. The odds —given this probability —are 9:1. 7. The bet is not even, although it may initially seem like it is. Adding up all the possible combinations of outcomes for the three rolled dice, it turns out that more than half are combinations not containing a six.
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It would be difficult to calculate all the combinations that do contain a six, but it’s much easier to figure out how many do not : each die would need to turn up with a number other than six, and each die has a 5/6 probability of doing that. So, for each die to come up with a non-six, the probability would be 5/6 x 5/6 x 5/6 = 125/216. This would be the probability of “losing.” The probability of “winning” is found via the Negation Rule: 1 – 125/216 = 91/216, which is less than a 50% chance. Janet was right to be cautious. # Combining the Rules We’ve already combined use of more than one rule. We did so repeatedly by using a conjunction or disjunction rule to determine the probability of an event not taking place, and then using the Negation Rule to determine the probability that it would. Often, we’ll need to use one or more rules multiple times to make a final calculation. We might have cause to determine the 221 probability of a conjunction and a second conjunction, a conjunction and a disjunction, a disjunction or another disjunction, and so on. We thus might use a combination of RCR, GCR, RDR, and GDR. We might even do that to lead up to a use of the Negation Rule. Just think of the fun! Let’s look at some examples. What is the probability of drawing two aces with replacement from a deck of 52 cards and rolling a two with a six-sided die? To determine the probability of independently drawing two aces, we use RCR: 4/52 x 4/52 = 1/13 x 1/13 = 1/ 169. To determine the probability of rolling a two, we simply use the Classical Theory: 1/6. To determine the probability of
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getting both , we use RCR (since drawing the aces and rolling the two are independent): 1/169 x 1/6 = 1/1014. What is the probability of rolling an even number with a six-sided die and getting at least one red card given two draws with replacement of the first card before the second draw? There are three “winners” for the die roll, so the probability is 3/6 = 1/2. We will draw at least one red card as long as we don’t draw two black cards. Let’s calculate that using RCR (the first card drawn is replaced, so the second result is independent of the first): 1/2 x 1/2 = 1/4. We now use the Negation Rule to calculate the probability of getting at least one red card: 1 – 1/4 = 3/4. Finally, we use RCR to determine the probability of both events occurring: 1/2 x 3/4 = 3/8. For the next three examples, consider the following scenario. Philosophy department peers of the three men and five women attending a regional symbolic logic conference agree on the following odds regarding what will happen to their close, conference-attending friends: the odds are 3:2 that Sarah will win a prize for best paper presented at the conference, 3:1 that Wanda will fall hopelessly in love with an exotic male dancer, 5:1 that Janet will want to play poker with some of the dancers, 1:2 that Betty will want to talk about modal logic, 1:3 that Nancy will want to quit her job teaching and become a restaurant chef, 7:2 that Mateo will ask Betty to work with him to publish an article about logic, 1:4 that Craig will sleep though the second half of the conference, and 4:1 that Pedro will try to convince the seven others that he should be
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the next department chair. (i) What is the probability that either Sarah will win a prize for best paper presented at the conference or Wanda will fall in love with an exotic male dancer, and both Craig will sleep through the second half of the conference and Mateo will ask Betty to work with him to publish an article about logic? First, we translate the odds to probabilities. For the events pertaining to Sarah, Wanda, Craig, and Mateo respectively, we get 3/5, 3/4, 1/5, and 7/9. Next we calculate the probability of the Sarah-or -Wanda events. Both might happen and one does not depend on the other, so we’ll use GDR: 3/5 + 3/4 – (3/5 x 3/4) = 27/20 – 9/20 = 18/20 = 9/10. Next we calculate the probability of the Craig-and -Mateo events using RCR (because one event has no impact on the other): 1/5 x 7/9 = 7/45. Finally, we want the probability of both of these results taking place, and since they are independent of each other, we’ll use RCR: 9/10 x 7/45 = 63/450. (ii) What is the probability of all the guy events occurring, and the Wanda or Betty event? 222 What we have here is a big, lumbering conjunction problem. The italicized and tell us that. Since the conjuncts are independent of each other, we’ll make our final calculation using RCR. For all the guy events happening we’ll use RCR. Since the Wanda and Betty events are not exc lusive, we’ll use GDR to determine the probability of either (or both) taking place. First, of course, we convert the accepted odds into probabilities. This is so much fun! For the guy events, we use RCR to get 7/9 x 1/5 x 4/5 = 28/225. For the Wanda-or-Betty events, we use GDR to
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get 3/4 + 1/3 – (3/4 x 1/3) = 9/12 + 4/12 – 3/12 = 10/12 = 5/6. Using RCR to get the probability of the conjunction of these two results, we get 28/225 x 5/6 = 140/1350 = 14/135. (iii) What is the probability that the Janet and Nancy events will both take place, or that the Craig and Pedro events will both take place? The Janet and Nancy events are independent, so we can use RCR to determine the probability of both occurring. So too with the Craig and Pedro events. Once we get each pair sorted out and a probability for each conjunction, we’ll use a disjunction rule to determine t he probability of either result occurring. Since both results can happen (they are independent and not mutually inclusive), we need to use GDR. The whole process will look like this: [P(J) x P(N)] + [P(C) x P(P)] – {[P(J) x P(N)] x [P(C) x P(P)]} Yuk. Let’s work through it, though. Determining the probabilities from the odds for each, we plug them in for the Janet, Nancy, Craig, and Pedro events, and get this: (5/6 x 1/4) + (1/5 x 4/5) – [(5/6 x 1/4) x (1/5 x 4/5)] Allowing our pre-college math skills to kick in, we are looking at… 5/24 + 4/25 – (5/24 x 4/25) = 9/25 – 20/600 = 216/600 – 20/600 = 196/600 = 49/150 # **Practice Problems: Combining Probability Rules Determine the probabilities called for in each problem below. You will likely need to use a combination of rules in each case. 1. What is the probability of getting two heads on two tosses of a coin, and drawing a red jack or a black card on one blind draw from deck of 52 cards? 2. What is the probability
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of rolling a number less than three on one roll of a normal six-sided die and getting heads on two honest tosses of coin? What are the odds? 3. What is the probability of getting at least one six on a roll of two six-sided dice, and drawing a queen from a deck of 52 cards? 4. Imagine two urns. The first urn contains two red balls, one green ball, and two yellow balls. The second urn contains three orange balls, two blue balls, and five purple balls. When drawing one ball from each urn, what is the probability of selecting either a red or green ball from the first urn, and either an orange or blue ball from the second urn? 223 5. Consider again the two urns from problem #4. Draw two balls from each urn. What is the probability of selecting red and yellow balls ( with replacement of the first ball) from the first urn, or orange and purple balls ( without replacement of the first ball) from the second urn? 6. Consider once again the two urns in problem # 4. Draw one ball from the first urn and two balls ( with replacement) from the second urn. What is the probability of selecting a green or yellow ball from the first urn, and both orange and purple balls from the second urn? 7. Yet again, consider the two urns in problem #4. Draw two balls from each of the urns. What is the probability of selecting either a red or green ball or both from the first urn ( without replacement), or either a blue or purple ball or both from the second urn ( without replacement)? Answers: 1. (1/2 x 1/2) x (2/52 + 26/52) = 1/4 x 28/52 = 1/4 x 7/13
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= 7/52 2. 2/6 x 1/2 x 1/2 = 1/12; 1:11 3. [(1/6 + 1/6) – (1/6 x1/6)] x 4/52 = 11/36 x 1/13 = 11/468 4. For the draw from the first urn, we use RDR: 2/5 + 1/5 = 3/5. For the draw from the second urn, we use the RDR: 3/10 + 2/10 = 5/10. To determine the probability of the conjunction of these to independent results, we use RCR: 3/5 x 5/10 = 15/50 = 3/10. 5. For the independent draws from the first urn, we use RCR: 2/5 x 2/5 = 4/25. For the draws (the second of which is dependent on the first) from the second urn, we use GCR: 3/10 x 5/9 = 15/90 = 1/6. Next we use GDR to determine the result of the independent and inclusive disjunction of these two results: 4/25 + 1/6 – (4/25 x 1/6) = 24/150 + 25/150 – 4/150 = 49/150 – 4/150 = 45/150 = 3/10. 6. For the draw from the first urn, we use the RDR: 1/5 + 2/5 = 3/5. For the draw from the second urn, we use the RCR: 3/10x 5/10 = 15/100 = 3/10. To determine the probability of the conjunction of the two independent results, we use RCR: 3/5 x 3/10 = 9/50. 7. For both pairs of urn draws, we face a second draw that is dependent on the first. Since these are disjunction calculations, they would be quite complex. So, instead, let’s determine the probability of losing on each draw, and then use the Negation Rule in each case to determine the probability of “winning” each draw. We’ll conclude by using GDR to determine the probably of “winning” with the first, or second (or both) urns. For the first urn scenario, th e only way
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to “lose” is to draw two yellow balls, so we’ll use GCR (we do not replace the first ball drawn): 2/5 x 1/4 = 2/20 = 1/10. To determine the probability of “winning” regarding the first urn, we now use the Negation Rule: 1 – 1/10 = 9/10. For the second urn scenario, the only way to “lose” is to draw two orange balls, so again we’ll use the GCR (because, again, we do not replace the first ball drawn before making the second draw): 3/10 x 2/9 = 6/90 = 2/45. Now we use the Negation Rule to determine the probability of winning”: 1 - 2/45 = 43/45. To win overall, we need to win with either the first urn, or the second urn, or both. Thus we use the GDR with the “winning” results of each pair of urn draws in mind: 9/10 + 2/45 – (9/10 x 2/45) = 405/450 + 20/450 – 18/450 = 407/450. # **Practice Problems: More Probability Calculations Use the Classical Theory, Relative Frequency Theory, Subjectivist Theory, or any combination of probability rules to perform the following calculations. Answers can be a fraction or a decimal number. 224 1. What is the probability of drawing a face card from a deck of 52 playing cards? 2. We observe 250 men go into a gym, and 12 of them use the rowing machine. What is the probability that a man entering the gym will use the rowing machine? 3. What is the probability —given the observations from problem #2, that a man entering the gym will not use the rowing machine? 4. The people most knowledgeable about the Seattle Mariners’ chances of winning this year’s World Series give them 1:25 odds of doing so. On this basis, what is the probability of their winning
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the upcoming World Series? 5. Consider an urn with two red balls and three green balls. What is the probability of selecting a white ball on one blind draw? 6. What is the probability of getting an ace or a jack from a deck of cards on one blind draw? 7. What is the probability of getting at least one ace on two draws from a deck of cards when the first card is replaced before the second is drawn? 8. What is the probability of getting two kings on two draws from a deck of cards, without replacement after the first draw? 9. What is the probability of getting a face card on a single draw from a deck of cards? 10. What is the probability of getting an even number or a three on any one of three rolls of a single die? 11. The odds are 2:3 that Bill will get an A on his English test. The odds are 2:1 that Sue will get an A on her English test. What is the probability that both Bill and Sue will get As on their English tests? 12. Refer to problem #6. What is the probability that either Bill or Sue (or both) will get an A on the test? 13. What is the probability of getting at least one tails on six tosses of a coin? 14. In a study of 250 baseball players, five developed severe elbow problems. In a study of 500 baseball players, ten developed a bone spur. What is the probability —based on these two studies —of a baseball player developing both severe elbow problems and a bone spur? 15. What is the probability of selecting at least one red ball on two draws from an urn containing two red balls, three
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white balls, and two green balls, when the first ball is replaced before the second selection? 16. Given the urn in problem #10, what is the probability of selecting either a red or a white ball (or both) on either of two draws, when the first ball is not replaced before the second draw? 17. Imagine two urns. The first urn contains three blue balls, one yellow ball, and two purple balls; the second urn contains four blue balls, two green balls, and one purple ball. You draw two balls from each urn, but must replace the first ball drawn from the first urn before the second draw; and you must not replace the first ball drawn from the second urn before the second draw. What is the probability of drawing a total of four blue balls? 18. Imagine the pair of urns in problem #12. What is the probability of drawing either a blue or yellow ball from the first urn on one blind draw, or of drawing two green balls in a row from the second urn when you replace the first green ball drawn before making the second draw into the second urn? 19. Imagine the pair of urns in problem #12. You get one blind draw into each. (a) What is the probability of your drawing a pink ball from the first urn or a green ball from the second? (b) What would be the probability if that question contained “and” instead of “or”? 225 20. Philosophy professors Wanda and Janet meet with Senator Sunny Shine to urge her to support state funding for student loans. They have known Sunny for years, working with her on gardening projects and activist bike rides. They believe the odds are 3:1 that Sunny will agree with them. There is
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also a 1/2 chance that Sunny will invite them to go to the beach with her that afternoon. What are the odds that Sunny will support funding for student loans and invite Wanda and Janet to go to the beach with her? Answers: 1. Use the Classical Theory. There are 12 face cards in a deck of 52: 12/52 = 3/13. 2. Using the Relative Frequency Theory, we get 12/250 = 6/125 or 0.048. 3. If the probability of a man using the rowing machine is 6/125, then the probability of his not using it will be 1 – 6/125 = 119/125 or 0.952. 4. 1/26 or 0.38 5. 0/5 = 0 (since there are no white balls in the urn, it’s impossible to draw one) 6. 2/13 or 0.15 7. 25/169 or 0.15 8. 1/221 or 0.0045 9. 3/13 or 0.23 10. 26/27 or 0.96 11. 4/15 or 0.27 12. 4/5 or 0.8 13. 63/64 or 0.98 14. 1/2500 or 0.0004 15. 24/49 or 0.49 16. 20/21 or 0.95 17. 1/14 or 0.071 18. 102/147 or 0.69 19. (a) 2/7 or 0.29, (b) 0 20. Odds of 3:1 convert to a probability of 3/4. Supporting student loans and inviting people to a beach are independent events, so we can use the RCR: 3/4 x 1/2 = 3/8 or 0.375. The odds would thus be 3:5.
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Title: [WP] This is the prologue (or the first chapter) of the novel you've always wanted to write. : r/WritingPrompts URL Source: Markdown Content: [WP] This is the prologue (or the first chapter) of the novel you've always wanted to write. : r/WritingPrompts =============== Skip to main content of the novel you've always wanted to write. : r/WritingPrompts Open menu Open navigation[]( to Reddit Home r/WritingPrompts A chip A close button Get App Get the Reddit app Log In of the novel you've always wanted to write. ============================================================================================ Writing Prompt EDIT 2: What the actual hell. Waking up to find your inbox at fifty - _and counting_ - is not healthy. Ya'lls are _machines_! EDIT 3: Does anybody here know what this "sleep" thing is? Cause I definitely don't. What the christ, people. Chill. Read more Share Share Add a comment Sort by: Best Open comment sort options * Best * Top * New * Controversial * Old * Q&A [![Image 6: u/KamikazeTomato avatar](
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[KamikazeTomato]( •[10y ago]( I was fourteen the day I realized I wasn’t a main character. It was the weather, you see. In the stories you could always count on the sky to get a good read on the situation. Thunder heavy in moments of dramatic crisis, shifts to rain for the emotionally sundering, and sunny and cloudless on days of jubilant celebration. The sky was like a heavenly emotional barometer, keenly tuned to the hero’s every passing mood. As for me, it was muggy and dreary the day I was born, raining by the bucketload when I graduated middle school, and sunny and chipper as anything—the morning my mother died. It was then in the cemetery, squinting up at that cheerful sun, that I realized. Life had displayed before me its glorious indifference. I was not special, I was not going to be much of anything, and then, eventually, I was going to die. It was with such thoughts that I entered high school, comfortable in the knowledge of my own insignificance. When something happened that changed my life forever. Reply reply } Share Share []( [THEDUDE33]( •[10y ago]( I think everything was good except for the last line. Too cliche for me. Reply reply } Share Share 8 more replies 8 more replies [More replies]( [![Image 7: u/thepiedpiper avatar]( [thepiedpiper]( •[10y ago]( "The Procrastinator's Guide to Getting Shit Done" Chapter One: I'll finish it later. Reply reply } Share Share [![Image 8: u/quantumfirefly avatar]( [quantumfirefly]( •[10y ago]( Remind me to call copyright infringement on this later. Reply reply } Share Share [![Image 9: u/quantumfirefly avatar]( [Cogh]( •[10y ago]( Reminds me of the [Procrastination Rock]( Reply reply } Share Share 2 more replies 2 more replies [More replies]( replies]( [![Image 10: u/ivangrozny avatar]( [ivangrozny]( •[10y ago]( Edited 10y ago This
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prompt is like a message from on high telling me to finally get started on the prologue idea I've been kicking around for my in-progress novel. Critiques are welcome. * * * Every fool and child knows that crossroads are magical places. Places where one might encounter things not easily found elsewhere. Where the desperate might cut a deal with a demon, and where the world-weary might find themselves crossing into another place or time. Every fool and child knows these things, but sometimes the wise forget them. The crossroads in question did not look like a crossroads to most who saw it. It looked like a simple dirt path, bisecting a field full of long grass that waved in the wind, the Earth's flowing green hair. On the path, at a seemingly random spot, there was a little town. But the townspeople could see the other half of the crossroads, as could the few who traveled upon it. The dirt path led from nowhere to nowhere, and the other road, which was made of something far less substantial than earth, led to just about any destination one could think of. And an inn had sprung up where the two roads met, as inns are wont to do. Later, a little town had built itself around the inn. The town was just like any other town, and the inn like any other. Horses brayed in their stables. Fields of crops grew. But if you were observant, and you walked along the wheat, you would notice something strange about it. It was. . . Wispy, almost. Shimmering. Almost like it was not quite real enough, or perhaps a bit too real. And if you walked through the horses' stables, or among the cows in the field, you would see a fierce,
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wild intelligence in the animals' eyes. The inn at the crossroads was called, simply, the Inn. And beside the hearth in its cozy taproom sat a man. This was true in a general sense, though the man often traveled around the town. But no matter where he was, if someone were to call on him, or even just walk into the Inn at an unexpected hour, they would always find him there, sitting by the hearth. The man was Storyteller, and every night he earned his name. The crowd in the Inn was always just right for the story at hand. No more than twelve people and no fewer than sixteen. The faces changed, but there were always a few familiar ones. Storyteller had his own names for the townspeople. There was Thinker, who sat in the back of the room with a mug of ale in his lap and a pensive expression on his face. Ploughman, a tough, hard-working farmer whose harvests always seemed poor, but who ploughed on nonetheless. His brother Harvester, who, unlike his kinsman, could sow one seed and get three plants. And Barkeep, of course, who kept everything in order at their little Inn. And then there was Storyteller's favorite audience member, a little girl he called Wonder. The only little girl in town, as a matter of fact. She always came to hear his stories. Tonight she seemed especially eager. "Are you ready? Are you ready yet?" "Almost, little one." Storyteller gave her a smile. He stood up and gave a dramatic cough, as was his custom. The room fell silent but for Wonder, who tried to say something else but was unable to make any sound but a squeal of delight. Storyteller turned his smile again to her, but this time it was
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tinged with sadness. In moments the girl, and everyone else in the Inn, would be trapped by his story. Their minds enslaved, unable to leave even figuratively, unable to think of anything else. Storyteller didn't think anyone could perceive this but himself, and still, it pained him greatly. He tried to at least spin his stories in a way that was worthy of this power, but who could be worthy of such terrible control? If only he could stop the telling. But no, he had tried. . . Storyteller reflected sadly on these things for a moment, as he often did, and then Wonder's voice pulled him back. "What's it about, tonight?" "Oh, you're in for a treat. It's a new one. I have a whole set of them for you. The Tales of the Frostlands, I'm calling them." "Oooooh," the little girl dragged the word out earnestly but absurdly, her eyes full of light. "_Who_ is it about?" "Well. . . It's about a lot of people, I suppose. But I'm in it," Storyteller grinned, "I'm my favorite character, in fact. Are you ready?" Wonder simply nodded. "Then listen. Listen." * * * * * * **Edit:** I hear you all loud and clear in re: to changing 'more' and 'fewer'. I give a short explanation in the comments below as to why I wrote it that way. That said, I can see that it clearly doesn't work as intended for most (all?) readers even when in context. I'll work on that. Edit 2: I really want to thank everyone individually, but at the same time I don't want to go crazy in the replies below. So if you gave me positive feedback or constructive criticism and I didn't thank you, I'll do so now. Thank you. Later today,
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maybe much later but definitely today, I'll have Chapter 1 posted on my subreddit, which is the same as my username ([r/ivangrozny]( Reply reply } Share Share [![Image 11: u/quantumfirefly avatar]( [quantumfirefly]( •[10y ago]( Glad I could help! Loved this, I want to know more about Storyteller! Just one criticism, sorry, the use of second person to explore a setting > And if you walked through the horses' stables, or among the cows in the field, you would see a fierce, wild intelligence in the animals' eyes. is a little jarring to me when combined immediately with the third person narration. But a great read! Reply reply } Share Share 15 more replies 15 more replies [More replies]( 43 more replies 43 more replies [More replies]( [![Image 12: u/see_mohn avatar]( [see_mohn]( •[10y ago]( I used to not believe in ghosts. I also used to love peanut butter. Now, I believe in ghosts and merely like peanut butter. I had a neighbor, you see. We went to different colleges, but hung out a lot when off. Not dating or anything, just to make it clear. Might have at some point, but we hadn't broached the subject. A month ago, both our schools had a long weekend. I forget why. Anyway, I stayed at school for the weekend because I was due to graduate soon. Today, in fact. I decided to enjoy my time at school while I could. She went home. Two days later, I got a call. She had died. Allergic reaction to peanut oil that had been in her lunch. I didn't touch peanut butter for a few days. Eventually, I caved. If you're not allergic to peanuts and don't like peanut butter, I have no words for you. As I mentioned, I graduated today. My parents and I
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got home in the evening, and I was too tired to bring any of my stuff up immediately. I decided to take a snack up and just bring my backpack with the electronics for now. I made a peanut butter sandwich. Hey, it's quick. I opened my door to see her, translucent and very clearly a ghost, on my couch. I'm very glad I was wearing my backpack at the time, because I totally would have dropped it. I had the sandwich in my mouth. Basically, I just walked in on a ghost while holding the murder weapon. This is going to get worse before it gets better. Reply reply } Share Share [![Image 13: u/quantumfirefly avatar]( [quantumfirefly]( •[10y ago]( If you live in the US, you better keep going. 'Cause we got this thing called the Constitution, which doesn't allow cruel and unusual punishment. Which means YOU CAN'T STOP. Reply reply } Share Share 9 more replies 9 more replies [More replies]( 11 more replies 11 more replies [More replies]( [![Image 14: u/IambWhatIamb avatar]( [IambWhatIamb]( •[10y ago]( It was the lunch rush and the PBJ Cafe was alive with voices. Each table's conversation fed into a swelling sound, joining the hiss of the espresso machine and the tinkle of plates and cups to create an effect that made Patrick think of running water. The stream of noise carried him in nearly constant motion as he wove between tables carrying sandwiches and lattes, and he imagined that he was steering around rocks in river rapids. While he flowed about the restaurant during busy stretches like this hours would pass like minutes, as if the volume of voices and the passage of time were somehow linked. The noise eventually ebbed as things slowed down, and with the afternoon lull setting in
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the staff complained about the quiet times. This was the kind of harmless shared suffering that strengthens social ties better than any team building exercise could and everyone joined in the commiserations. They would have complained about the busy times too if they didn't have so many other things to take care of during the rush. But when things slowed down the tips stopped coming, the minutes dragged by, and boredom took hold. Patrick agreed, yet he secretly enjoyed it when the cafe quieted down because the voices at the tables would once again separate into distinct conversations that he could follow. While the other staff were smoking out back or inventing elaborate games in the store room around throwing butter knives into the drywall, Patrick was collecting stories. Sometimes he felt guilty about eavesdropping, but from the central counter it was possible to hear what was being said almost anywhere in the cafe, and the gap between hearing and listening is so small that we often cross it without realizing. Patrick first began to cross that gap unintentionally because he was worried that the people were talking about him. He was a novice waiter and felt that his inexperience must have been obvious to the customers. However, as he listened in on his tables it eventually sunk in that they were not, in fact, discussing him at all. It turned out that people spent less time thinking about and talking about him than he was prone to imagine. He came to see that paranoia was just as self-centered and deluded as narcissism, without the benefit of confidence. This realization, coupled with his growing competence at the job, helped him to stop worrying that the customers were criticizing him. But not before weeks of eavesdropping had also taught him that
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{
"page_id": null,
"source": 6820,
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people said some interesting things in restaurants. Granted, people said a lot of very boring things in restaurants. As well as a huge number of things that, lacking context, Patrick couldn't really gauge one way or the other. But there were enough intriguing moments to keep him coming back. The first was a woman with thick, dirty blonde hair discussing her nervous breakdown in such unguarded detail and with so little appeal for sympathy that Patrick fell in love with her a little bit, though she was twice his age. He was in awe of that kind of openness, especially about such a moment of weakness. But the man in glasses seated across from her did not seem impressed. Maybe he'd heard the story before, or maybe this was a first date and he was having second thoughts. He could have been her shrink as well, though he wasn’t taking notes or asking many questions. Later that same day Patrick overheard a young white guy professing his love to a young black guy with such whispered urgency it seemed he had to keep his voice down so as not to shout. Patrick felt the urge to hug them both, and was only a little afraid that this might mean he was a homosexual. As he set their drinks down in front of them he wanted to tell them that he supported gay marriage, and interracial marriage, and any kind of marriage really if it involved a love such as theirs. All he said, though, was to just let him know if they needed anything else. From then on he was hooked. When he had down time he would busy himself behind the counter and tune in to the different conversations going on in the restaurant around him. He justified his
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{
"page_id": null,
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eavesdropping by thinking that the PBJ Cafe was clearly a public place, so people should expect to be overheard. Sometimes he took things a step further and went the righteous route: if the customers only thought of him as a server and not as a fully formed human being, capable of hearing and maybe even having opinions about what they were saying, then he had every right to listen to them with no qualms. Eavesdropping as a form of social justice was a difficult concept to hold onto on this particular afternoon, though. The problem was: she was cute. Reply reply } Share Share 8 more replies 8 more replies [More replies]( New to Reddit? Create your account and connect with a world of communities. Continue with Email Continue With Phone Number By continuing, you agree to our[User Agreement]( acknowledge that you understand the[Privacy Policy]( More posts you may like ======================= * [[WP] The hero was killed by the villain in the stupidest way imaginable and everyone, even the villain, is convinced that they must have staged their death. A month has passed and the realisation that the hero really is dead is slowly setting in.]( 15: r/WritingPrompts icon]( r/WritingPrompts]( mo. ago ![Image 16: A banner for the subreddit]( ![Image 17: r/WritingPrompts icon]( Writing Prompts. You're a writer and you just want to flex those muscles? You've come to the right place! If you see a prompt you like, simply write a short story based on it. Get comments from others, and leave commentary for other people's works. Let's help each other. * * * 19M Members Online [### [WP] The hero was killed by the villain in the stupidest way imaginable and everyone, even the villain, is convinced that they must have staged their death. A month has passed
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{
"page_id": null,
"source": 6820,
"title": "from dpo"
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and the realisation that the hero really is dead is slowly setting in.]( 135 upvotes ·11 comments * * * * [[WP] The hero has finally confronted you, the villain, for killing their parents all those years ago. You try to act like you know, but you really don't recall who they or their parents are.]( 18: r/WritingPrompts icon]( r/WritingPrompts]( mo. ago ![Image 19: A banner for the subreddit]( ![Image 20: r/WritingPrompts icon]( Writing Prompts. You're a writer and you just want to flex those muscles? You've come to the right place! If you see a prompt you like, simply write a short story based on it. Get comments from others, and leave commentary for other people's works. Let's help each other. * * * 19M Members Online [### [WP] The hero has finally confronted you, the villain, for killing their parents all those years ago. You try to act like you know, but you really don't recall who they or their parents are.]( 71 upvotes ·12 comments * * * * [[WP] A fictional character comes to life and meets their creator, but, rather than raging against their maker, they're more interested in what inspired them]( 21: r/WritingPrompts icon]( r/WritingPrompts]( mo. ago ![Image 22: A banner for the subreddit]( ![Image 23: r/WritingPrompts icon]( Writing Prompts. You're a writer and you just want to flex those muscles? You've come to the right place! If you see a prompt you like, simply write a short story based on it. Get comments from others, and leave commentary for other people's works. Let's help each other. * * * 19M Members Online [### [WP] A fictional character comes to life and meets their creator, but, rather than raging against their maker, they're more interested in what inspired them]( 40 upvotes ·10 comments *
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{
"page_id": null,
"source": 6820,
"title": "from dpo"
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* * * Promoted ![Image 24: sidebar promoted post thumbnail]( For as long as history remembers, prophecies has ruled the world, heroes bathed in glory, villains drowned in infamy. And people like you? Always caught in between. Destined to be nothing more than pawns in a unavoidable script. So you set out to destroy the ultimate evil: Fate itself.]( 25: r/WritingPrompts icon]( r/WritingPrompts]( mo. ago ![Image 26: A banner for the subreddit]( ![Image 27: r/WritingPrompts icon]( Writing Prompts. You're a writer and you just want to flex those muscles? You've come to the right place! If you see a prompt you like, simply write a short story based on it. Get comments from others, and leave commentary for other people's works. Let's help each other. * * * 19M Members Online [### [WP] For as long as history remembers, prophecies has ruled the world, heroes bathed in glory, villains drowned in infamy. And people like you? Always caught in between. Destined to be nothing more than pawns in a unavoidable script. So you set out to destroy the ultimate evil: Fate itself.]( 61 upvotes ·12 comments * * * * [[WP] As an immortal, everyone you love eventually dies. So instead of looking for a singular sapient companion, you resolve to create a work of fiction so popular that it's community shall live alongside you for thousands of years.]( 28: r/WritingPrompts icon]( r/WritingPrompts]( mo. ago ![Image 29: A banner for the subreddit]( ![Image 30: r/WritingPrompts icon]( Writing Prompts. You're a writer and you just want to flex those muscles? You've come to the right place! If you see a prompt you like, simply write a short story based on it. Get comments from others, and leave commentary for other people's works. Let's help each other. * * * 19M Members
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{
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"source": 6820,
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Online [### [WP] As an immortal, everyone you love eventually dies. So instead of looking for a singular sapient companion, you resolve to create a work of fiction so popular that it's community shall live alongside you for thousands of years.]( 382 upvotes ·19 comments * * * * [[WP] Your a teacher who got summoned to another world with your students. Your about to be kicked out by the king for being the only one without powers, but you beg to stay because, "letting a bunch of emotional teens with god powers run around without any sort of guidance is a terrible idea."]( 31: r/WritingPrompts icon]( r/WritingPrompts]( days ago ![Image 32: A banner for the subreddit]( ![Image 33: r/WritingPrompts icon]( Writing Prompts. You're a writer and you just want to flex those muscles? You've come to the right place! If you see a prompt you like, simply write a short story based on it. Get comments from others, and leave commentary for other people's works. Let's help each other. * * * 19M Members Online [### [WP] Your a teacher who got summoned to another world with your students. Your about to be kicked out by the king for being the only one without powers, but you beg to stay because, "letting a bunch of emotional teens with god powers run around without any sort of guidance is a terrible idea."]( 388 upvotes ·19 comments * * * * [[WP] Long ago, be either from a bioweapon or an ancient curse all of humanity transformed into mindless monsters. The world fell, and the creatures wandered for centuries. But now, something is changing. The monsters are starting to remember who they were.]( 34: r/WritingPrompts icon]( r/WritingPrompts]( days ago ![Image 35: A banner for the subreddit]( ![Image 36: r/WritingPrompts icon]( Writing
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{
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"source": 6820,
"title": "from dpo"
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Prompts. You're a writer and you just want to flex those muscles? You've come to the right place! If you see a prompt you like, simply write a short story based on it. Get comments from others, and leave commentary for other people's works. Let's help each other. * * * 19M Members Online [### [WP] Long ago, be either from a bioweapon or an ancient curse all of humanity transformed into mindless monsters. The world fell, and the creatures wandered for centuries. But now, something is changing. The monsters are starting to remember who they were.]( 101 upvotes ·17 comments * * * * [[WP] "Here to kill me, heroes? Fools! I will never submit!" "No, actually." "What?" "We're actually here to see if you'll help US conquer the world. If not, THEN we kill you, but right now it's up to you what happens."]( 37: r/WritingPrompts icon]( r/WritingPrompts]( mo. ago ![Image 38: A banner for the subreddit]( ![Image 39: r/WritingPrompts icon]( Writing Prompts. You're a writer and you just want to flex those muscles? You've come to the right place! If you see a prompt you like, simply write a short story based on it. Get comments from others, and leave commentary for other people's works. Let's help each other. * * * 19M Members Online [### [WP] "Here to kill me, heroes? Fools! I will never submit!" "No, actually." "What?" "We're actually here to see if you'll help US conquer the world. If not, THEN we kill you, but right now it's up to you what happens."]( 65 upvotes ·16 comments * * * * Promoted ![Image 40: sidebar promoted post thumbnail]( For centuries, a legendary hero had fought and defended humanity from outsider threats, only to watch as men killed one another afterwards for the loot,
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{
"page_id": null,
"source": 6820,
"title": "from dpo"
}
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now, a new outsider threat emerges, and the hero only has one thing to say.]( 41: r/WritingPrompts icon]( r/WritingPrompts]( mo. ago ![Image 42: A banner for the subreddit]( ![Image 43: r/WritingPrompts icon]( Writing Prompts. You're a writer and you just want to flex those muscles? You've come to the right place! If you see a prompt you like, simply write a short story based on it. Get comments from others, and leave commentary for other people's works. Let's help each other. * * * 19M Members Online [### [WP] For centuries, a legendary hero had fought and defended humanity from outsider threats, only to watch as men killed one another afterwards for the loot, now, a new outsider threat emerges, and the hero only has one thing to say.]( 125 upvotes ·9 comments * * * * [[WP] When you were offered the opportunity to save people across countless worlds, you took it. Countless worlds later, you're jaded, bitter, traumatized, and cursed, but you still save people. It's the only thing you have left.]( 44: r/WritingPrompts icon]( r/WritingPrompts]( mo. ago ![Image 45: A banner for the subreddit]( ![Image 46: r/WritingPrompts icon]( Writing Prompts. You're a writer and you just want to flex those muscles? You've come to the right place! If you see a prompt you like, simply write a short story based on it. Get comments from others, and leave commentary for other people's works. Let's help each other. * * * 19M Members Online [### [WP] When you were offered the opportunity to save people across countless worlds, you took it. Countless worlds later, you're jaded, bitter, traumatized, and cursed, but you still save people. It's the only thing you have left.]( 140 upvotes ·9 comments * * * * [This infuriating word puzzle needs to be solved,
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{
"page_id": null,
"source": 6820,
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for the sake of the sanity of many.]( 47: r/puzzles icon]( r/puzzles]( yr. ago ![Image 48: r/puzzles icon]( The place for all kinds of puzzles including puzzle games. Self-promotion is allowed in the stickied "Promo Weekly" post. * * * 425K Members Online [### This infuriating word puzzle needs to be solved, for the sake of the sanity of many.]( [![Image 49: r/puzzles - This infuriating word puzzle needs to be solved, for the sake of the sanity of many.]( 576 upvotes ·506 comments * * * * [[WP] Many legends exist of heroes traveling to the underworld to save someone they love. You are not one of them. You went down there because you are not done with them.]( 50: r/WritingPrompts icon]( r/WritingPrompts]( days ago ![Image 51: A banner for the subreddit]( ![Image 52: r/WritingPrompts icon]( Writing Prompts. You're a writer and you just want to flex those muscles? You've come to the right place! If you see a prompt you like, simply write a short story based on it. Get comments from others, and leave commentary for other people's works. Let's help each other. * * * 19M Members Online [### [WP] Many legends exist of heroes traveling to the underworld to save someone they love. You are not one of them. You went down there because you are not done with them.]( 205 upvotes ·18 comments * * * * [[WP] You are a villain and you were recently possessed by a good spirit. Today you wake up, not remembering the last few days, to news of a new and mysterious hero that has appeared.]( 53: r/WritingPrompts icon]( r/WritingPrompts]( mo. ago ![Image 54: A banner for the subreddit]( ![Image 55: r/WritingPrompts icon]( Writing Prompts. You're a writer and you just want to flex those muscles? You've come
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{
"page_id": null,
"source": 6820,
"title": "from dpo"
}
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to the right place! If you see a prompt you like, simply write a short story based on it. Get comments from others, and leave commentary for other people's works. Let's help each other. * * * 19M Members Online [### [WP] You are a villain and you were recently possessed by a good spirit. Today you wake up, not remembering the last few days, to news of a new and mysterious hero that has appeared.]( 37 upvotes ·11 comments * * * * [Ending of My Hero Academia is so unrealistic]( 56: r/MyHeroAcadamia icon]( r/MyHeroAcadamia]( mo. ago ![Image 57: A banner for the subreddit]( ![Image 58: r/MyHeroAcadamia icon]( Welcome to a My Hero Academia Subreddit! A subreddit for the dub fans as well as sub fans. Everyone is welcome to post anime related content. Feel free to create or post drawings, edits/clips, or whatever you like from my hero or related anime. Emojis by @rennomiya on instagram and @小栗ミト on pixiv! * * * 171K Members Online [SPOILER ### Ending of My Hero Academia is so unrealistic]( 1.6K upvotes ·310 comments * * * * [What is one stoic quote that had the biggest impact on you?]( 59: r/Stoicism icon]( r/Stoicism]( yr. ago ![Image 60: r/Stoicism icon]( We are a community committed to learning about and applying philosophical Stoic principles and techniques. * * * 730K Members Online [### What is one stoic quote that had the biggest impact on you?]( 420 upvotes ·183 comments * * * * [The false promise of speed reading - why you should read slowly if you want to understand]( 61: r/books icon]( r/books]( yr. ago ![Image 62: A banner for the subreddit]( ![Image 63: r/books icon]( This is a moderated subreddit. It is our intent and purpose to foster and encourage in-depth
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"page_id": null,
"source": 6820,
"title": "from dpo"
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discussion about all things related to books, authors, genres, or publishing in a safe, supportive environment. If you're looking for help with a personal book recommendation, consult our Weekly Recommendation Thread, Suggested Reading page, or ask in r/suggestmeabook. * * * 27M Members Online ### The false promise of speed reading - why you should read slowly if you want to understand, drama reviews, official soundtracks, news, award shows and more. Be sure to
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"source": 6820,
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check out our sidebar for helpful info and resources! * * * 1.1M Members Online [### Family by Choice [Episodes 3 & 4]]( 176 upvotes ·209 comments * * * * [How to get a full body image?]( 73: r/StableDiffusion icon]( r/StableDiffusion]( yr. ago ![Image 74: A banner for the subreddit]( ![Image 75: r/StableDiffusion icon]( /r/StableDiffusion is an unofficial community embracing the open-source material of all related. Post art, ask questions, create discussions, contribute new tech, or browse the subreddit. It’s up to you. * * * 739K Members Online [### How to get a full body image?]( 19 upvotes ·22 comments * * * * [Raise wa Tanin ga Ii • Yakuza Fiancé: Raise wa Tanin ga Ii - Episode 11 discussion]( 76: r/anime icon]( r/anime]( mo. ago ![Image 77: r/anime icon]( Reddit's premier anime community. * * * 14M Members Online [### Raise wa Tanin ga Ii • Yakuza Fiancé: Raise wa Tanin ga Ii - Episode 11 discussion]( 284 upvotes ·101 comments * * * * [[WP] A desperate man makes a final appeal to the Gods before his demise. The Gods do not answer. Something else does.]( 78: r/WritingPrompts icon]( r/WritingPrompts]( mo. ago ![Image 79: A banner for the subreddit]( ![Image 80: r/WritingPrompts icon]( Writing Prompts. You're a writer and you just want to flex those muscles? You've come to the right place! If you see a prompt you like, simply write a short story based on it. Get comments from others, and leave commentary for other people's works. Let's help each other. * * * 19M Members Online [### [WP] A desperate man makes a final appeal to the Gods before his demise. The Gods do not answer. Something else does.]( 94 upvotes ·10 comments * * * * [[WP] One day, you get a
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{
"page_id": null,
"source": 6820,
"title": "from dpo"
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knock on your door. When you open it, you see the protagonist from your favorite show standing there, wide-eyed. "I know you won't believe me," they say, "but you're the main character of my favorite book. I know how it ends and I'm here to change it."]( 81: r/WritingPrompts icon]( r/WritingPrompts]( mo. ago ![Image 82: A banner for the subreddit]( ![Image 83: r/WritingPrompts icon]( Writing Prompts. You're a writer and you just want to flex those muscles? You've come to the right place! If you see a prompt you like, simply write a short story based on it. Get comments from others, and leave commentary for other people's works. Let's help each other. * * * 19M Members Online [### [WP] One day, you get a knock on your door. When you open it, you see the protagonist from your favorite show standing there, wide-eyed. "I know you won't believe me," they say, "but you're the main character of my favorite book. I know how it ends and I'm here to change it."]( 133 upvotes ·6 comments * * * * [The Book of Job Destroyed my Faith]( 84: r/Christianity icon]( r/Christianity]( yr. ago ![Image 85: A banner for the subreddit]( ![Image 86: r/Christianity icon]( /r/Christianity is a subreddit to discuss Christianity and aspects of Christian life. All are welcome to participate. * * * 544K Members Online [### The Book of Job Destroyed my Faith]( 82 upvotes ·385 comments * * * * [CMV: Soccer is inherently boring and hard to watch]( 87: r/changemyview icon]( r/changemyview]( yr. ago ![Image 88: r/changemyview icon]( A place to post an opinion you accept may be flawed, in an effort to understand other perspectives on the issue. Enter with a mindset for conversation, not debate. * * * 3.9M Members Online [### CMV:
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{
"page_id": null,
"source": 6820,
"title": "from dpo"
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{
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"title": "from dpo"
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* [Superhero Movies & Series]( * [TV News & Discussion]( * * * * RESOURCES * [About Reddit]( * [Advertise]( * [Reddit Pro BETA]( * [Help]( * [Blog]( * [Careers]( * [Press]( * * * * [Communities]( * [Best of Reddit]( * [Topics]( ![Image 91](
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{
"page_id": null,
"source": 6820,
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Title: Exemplar Texts for Grades 6-8 URL Source: Published Time: 2011-07-06T17:14:33.000Z Markdown Content: COMMON CORE STATE ST ANDARDS FOR # English Language Arts # & # Literacy in # History/Social Studies, # Science, and Technical Subje cts # _____ # Appendix B: # Text Exemplars and # Sample Performance Tasks APPENDIX B 2 > OREGON COMMON CORE STATE STANDARDS FOR English Language Arts > & Literacy in History/Social Studies, Science, and Technical Subjects # Exemplars of Reading Text Complexity, Quality, and Range & Sample Performance Tasks Related to Core Standards # Selecting Text Exemplars The following text samples primarily serve to exemplify the level of complexity and quality that the Standards require all students in a given grade band to engage with. Additionally, they are suggestive of the breadth of texts that students should encounter in the text types required by the Standards. The choices should serve as useful guideposts in helping educators select texts of similar complexity , quality ,and range for their own classrooms. They expressly do not represent a partial or complete reading list. The process of text selection was guided by the following criteria: Complexity. Appendix A describes in detail a three-part model of measuring text complexity based on qualitative and quantitative indices of inherent text difficulty balanced with educators’ professional judgment in matching readers and texts in light of particular tasks. In selecting texts to serve as exemplars, the work group began by soliciting contributions from teachers, educational leaders, and researchers who have experience working with students in the grades for which the texts have been selected. These contributors were asked to recommend texts that they or their colleagues have used successfully with students in a given grade band. The work group made final selections based in part on whether qualitative and quantitative measures
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{
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indicated that the recommended texts were of sufficient complexity for the grade band. For those types of texts —particularly poetry and multimedia sources —for which these measures are not as well suited, professional judgment necessarily played a greater role in selection. Quality. While it is possible to have high-complexity texts of low inherent quality, the work group solicited only texts of recognized value. From the pool of submissions gathered from outside contributors, the work group selected classic or historically significant texts as well as contemporary works of comparable literary merit, cultural significance, and rich content. Range. After identifying texts of appropriate complexity and quality, the work group applied other criteria to ensure that the samples presented in each band represented as broad a range of sufficiently complex, high-quality texts as possible. Among the factors considered were initial publication date, authorship, and subject matter. # Copyright and Permissions For those exemplar texts not in the public domain, we secured permissions and in some cases employed a conservative interpretation of fair use, which allows limited, partial use of copyrighted text for a nonprofit educational purpose as long as that purpose does not impair the rights holder’s ability to seek a fair return for his or her work. In instances where we could not employ fair use and have been unable to secure permission, we have listed a title without providing an excerpt. Thus, some short texts are not excerpted here, as even short passages from them would constitute a substantial portion of the entire work. In addition, illustrations and other graphics in texts are generally not reproduced here. Such visual elements are particularly important in texts for the youngest students and in many informational texts for readers of all ages. (Using the qualitative criteria outlined in Appendix A, the work group considered
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the importance and complexity of graphical elements when placing texts in bands.) APPENDIX B 3 > OREGON COMMON CORE STATE STANDARDS FOR English Language Arts > & Literacy in History/Social Studies, Science, and Technical Subjects When excerpts appear, they serve only as stand-ins for the full text. The Standards require that students engage with appropriately complex literary and informational works; such complexity is best found in whole texts rather than passages from such texts. Please note that these texts are included solely as exemplars in support of the Standards. Any additional use of those texts that are not in the public domain, such as for classroom use or curriculum development, requires independent permission from the rights holders. The texts may not be copied or distributed in any way other than as part of the overall Common Core State Standards Initiative documents. # Sample Performance Tasks The text exemplars are supplemented by brief performance tasks that further clarify the meaning of the Standards. These sample tasks illustrate specifically the application of the Standards to texts of sufficient complexity, quality, and range. Relevant Reading standards are noted in brackets following each task, and the words in italics in the task reflect the wording of the Reading standard itself. (Individual grade-specific reading standards are identified by their strand, grade, and number, so that RI.4.3, for example, stands for Reading, Informational Text, grade 4, standard 3.) # How to Read This Document The materials that follow are divided into text complexity grade bands as defined by the Standards: K –1, 2–3, 4 –5, 6 –8, 9 –10, and 11 –CCR. Each band’s exemplars are divided into text types matching those required in the Standards for a given grade. K –5 exemplars are separated into stories, poetry, and informational texts (as well as read-aloud texts
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{
"page_id": null,
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in kindergarten through grade 3). The 6 –CCR exemplars are divided into English language arts (ELA), history/social studies, and science, mathematics, and technical subjects, with the ELA texts further subdivided into stories, drama, poetry, and informational texts. (The history/social studies texts also include some arts-related texts.) Citations introduce each excerpt, and additional citations are included for texts not excerpted in the appendix. Within each grade band and after each text type, sample performance tasks are included for select texts. # Media Texts Selected excerpts are accompanied by annotated links to related media texts freely available online at the time of the publication of this document. APPENDIX B 4 > OREGON COMMON CORE STATE STANDARDS FOR English Language Arts > & Literacy in History/Social Studies, Science, and Technical Subjects # Table of Contents ## K–1 Text Exemplars ............................................................................................................. 14 ## Stories ............................................................................................................................. 14 Minarik, Else Holmelund. Little Bear . ................................................................................................. 14 Eastman, P. D. Are You My Mother? .................................................................................................. 15 Seuss, Dr. Green Eggs and Ham . ........................................................................................................ 15 Lopshire, Robert. Put Me in the Zoo . ................................................................................................. 16 Mayer, Mercer. A Boy, a Dog and a Frog . .......................................................................................... 16 Lobel, Arnold. Frog and Toad Together . ............................................................................................ 16 Lobel, Arnold. Owl at Home . .............................................................................................................. 17 DePaola, Tomie. Pancakes for Breakfast . .......................................................................................... 18 Arnold, Tedd. Hi! Fly Guy .................................................................................................................... 18 ## Poetry ............................................................................................................................. 19 Anonymous. “As I Was Going to St. Ives.” ......................................................................................... 19 Rossetti, Christina. “Mix a Pancake.” ................................................................................................. 19 Fyleman, Rose. “Singing -Time.” ......................................................................................................... 19 Milne, A. A. “Halfway Down.” ............................................................................................................ 19 Chute, Marchette. “Drinking Fountain.” ............................................................................................ 20 Hughes, Langston. “Poem.” ................................................................................................................ 20 Ciardi , John. “Wouldn’t You?” ............................................................................................................ 20 Wright, Richard. “Laughing Boy.” ....................................................................................................... 20 Greenfield, Eloise. “By Myself.” ......................................................................................................... 21 Giovanni, Nikki. “Covers.” .................................................................................................................. 21 Merriam, Eve. “It Fell in the City.” ...................................................................................................... 21
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Lopez, Alonzo. “Celebration.” ............................................................................................................ 21 Agee, Jon. “Two Tree Toads.” ............................................................................................................ 22 ## Read-Aloud Stories .......................................................................................................... 22 Baum, L. Frank. The Wonderful Wizard of Oz . ................................................................................... 22 Wilder, Laura Ingalls. Little House in the Big Woods . ......................................................................... 23 Atwater, Richard and Florence. Mr. Popper’s Penguins .................................................................... 24 Jansson, Tove. Finn Family Moomintroll ............................................................................................ 24 Haley, Gail E. A Story, A Story ............................................................................................................. 25 Bang, Molly. The Paper Crane ............................................................................................................ 26 Young, Ed. Lon Po Po: A Red-Riding Hood Story from China .............................................................. 27 Garza, Carmen Lomas. Family Pictures .............................................................................................. 27 Mora, Pat. Tomás and the Library Lady ............................................................................................. 28 Henkes, Kevin. Kitten’s First Full Moon .............................................................................................. 29 ## Read-Aloud Poetry .......................................................................................................... 30 Anonymous. “The Fox’s Foray.” ......................................................................................................... 30 Langstaff, John. Over in the Meadow ................................................................................................. 32 Lear, Edward. “The Owl and the Pussycat.” ....................................................................................... 32 Hughes, Langston. “April Rain Song.” ................................................................................................ 33 APPENDIX B 5 > OREGON COMMON CORE STATE STANDARDS FOR English Language Arts > & Literacy in History/Social Studies, Science, and Technical Subjects Moss, Lloyd. Zin! Zin! Zin! a Violin ...................................................................................................... 33 ## Sample Performance Tasks for Stories and Poetry ............................................................ 34 ## Informational Texts ......................................................................................................... 35 Bulla, Clyde Robert. A Tree Is a Plant . ................................................................................................ 35 Aliki. My Five Senses ........................................................................................................................... 36 Hurd, Edith Thacher. Starfish ............................................................................................................. 37 Aliki. A Weed is a Flower: The Life of George Washington Carver ..................................................... 37 Crews, Donald. Truck .......................................................................................................................... 38 Hoban, Tana. I Read Signs .................................................................................................................. 38 Reid, Mary Ebeltoft. Let’s Find Out About Ice Cream ......................................................................... 38 “Garden Helpers.” National Geographic Young Explorers ................................................................. 38 “Wind Power.” National Geographic Young Explorers ...................................................................... 38 ## Read-Aloud Informational Texts ...................................................................................... 39 Provensen, Alice and Martin. The Year at Maple Hill Farm . .............................................................. 39 Gibbons, Gail. Fire! Fire! ..................................................................................................................... 39 Dorros, Arthur. Follow the Water from Brook to
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Ocean . ................................................................... 40 Rauzon, Mark, and Cynthia Overbeck Bix. Water, Water Everywhere . ............................................. 41 Llewellyn, Claire. Earthworms . ........................................................................................................... 41 Jenkins, Steve, and Robin Page. What Do You Do With a Tail Like This? ........................................... 41 Pfeffer, Wendy. From Seed to Pumpkin . ............................................................................................ 42 Thomson, Sarah L. Amazing Whales! ................................................................................................. 42 Hodgkins, Fran, and True Kelley. How People Learned to Fly . ........................................................... 43 Nivola, Claire A. Planting the Trees of Kenya: The Story of Wangari Maathai . ................................ 45 ## Sample Performance Tasks for Informational Texts .......................................................... 45 ## Grades 2 –3 Text Exemplars .................................................................................................. 46 ## Stories ............................................................................................................................. 46 Gannett, Ruth Stiles. My Father’s Dragon ......................................................................................... 46 Averill, Esther. The Fire Cat . ............................................................................................................... 47 Steig, William. Amos & Boris . ............................................................................................................. 48 Shulevitz, Uri. The Treasure ................................................................................................................ 48 Cameron, Ann. The Stories Julian Tells . .............................................................................................. 48 MacLachlan, Patricia. Sarah, Plain and Tall ........................................................................................ 48 Rylant, Cynthia. Henry and Mudge: The First Book of Their Adventures . .......................................... 49 Stevens, Janet. Tops and Bottoms ...................................................................................................... 50 LaMarche, Jim. The Raft . .................................................................................................................... 51 Rylant, Cynthia. Poppleton in Winter . ................................................................................................ 51 Rylant, Cynthia. The Lighthouse Family : ............................................................................................ 52 Osborne, Mary Pope. The One-Eyed Giant (Book One of Tales from the Odyssey) . .......................... 53 Silverman, Erica. Cowgirl Kate and Cocoa . ......................................................................................... 54 ## Poetry ............................................................................................................................. 55 Dickinson, Emily. “Autumn.” .............................................................................................................. 55 Rossetti, Christina. “Who Has Seen the Wind?” ................................................................................ 55 Millay, Edna St. Vincent. “Afternoon on a Hill.” ................................................................................. 55 APPENDIX B 6 > OREGON COMMON CORE STATE STANDARDS FOR English Language Arts > & Literacy in History/Social Studies, Science, and Technical Subjects Frost, Robert. “Stopping by Woods on a Snowy Evening.” ................................................................ 56 Field, Rachel. “Something Told the Wild Geese.” .............................................................................. 56 Hughes, Langston. “Grandpa’s Stories.” ............................................................................................ 56 Jarrell, Randall. “A
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Bat Is Born.” ......................................................................................................... 56 Giovanni, Nikki. “Knoxville, Tennessee.” ............................................................................................ 57 Merriam, Eve. “Weather.” ................................................................................................................. 58 Soto, Gary. “Eating While Reading.” .................................................................................................. 58 ## Read-Aloud Stories .......................................................................................................... 58 Kipling, Rudyard. “How the Camel Got His Hump.” ........................................................................... 58 Thurber, James. The Thirteen Clocks . ................................................................................................. 60 White, E. B. Charlotte’s Web . ............................................................................................................. 60 Selden, George. The Cricket in Times Square . .................................................................................... 61 Babbitt, Natalie. The Search for Delicious . ......................................................................................... 62 Curtis, Christopher Paul. Bud, Not Buddy . ......................................................................................... 62 Say, Allen. The Sign Painter . ............................................................................................................... 64 ## Read-Aloud Poetry .......................................................................................................... 64 Lear, Edward. “The Jumblies.” ........................................................................................................... 64 Browning, Robert. The Pied Piper of Hamelin . ................................................................................... 66 Johnson, Georgia Douglas. “Your World.” ......................................................................................... 67 Eliot, T. S. “The Song of the Jellicles.” ................................................................................................ 68 Fleischman, Paul. “Fireflies.” .............................................................................................................. 68 ## Sample Performance Tasks for Stories and Poetry ............................................................ 68 ## Informational Texts ......................................................................................................... 69 Aliki. A Medieval Feast . ...................................................................................................................... 69 Gibbons, Gail. From Seed to Plant . ..................................................................................................... 70 Milton, Joyce. Bats: Creatures of the Night . ...................................................................................... 70 Beeler, Selby. Throw Your Tooth on the Roof: Tooth Traditions Around the World . ......................... 71 Leonard, Heather. Art Around the World . .......................................................................................... 71 Ruffin, Frances E. Martin Luther King and the March on Washington .............................................. 72 St. George, Judith. So You Want to Be President? ............................................................................. 72 Einspruch, Andrew. Crittercam . ......................................................................................................... 72 Kudlinski, Kathleen V. Boy, Were We Wrong About Dinosaurs . ........................................................ 72 Davies, Nicola. Bat Loves the Night . ................................................................................................... 73 Floca, Brian. Moonshot: The Flight of Apollo 11 . ............................................................................... 73 Thomson, Sarah L. Where Do Polar Bears Live? ................................................................................. 74 ## Read-Aloud Informational Texts ...................................................................................... 75 Freedman, Russell. Lincoln: A Photobiography . ................................................................................. 75 Coles, Robert. The Story of Ruby Bridges . .......................................................................................... 76 Wick, Walter. A Drop of
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Water: A Book of Science and Wonder . ...................................................... 77 Smith, David J. If the World Were a Village: A Book about the World’s People . ............................... 77 Aliki. Ah, Music! .................................................................................................................................. 78 Mark, Jan. The Museum Book: A Guide to Strange and Wonderful Collections ................................. 78 D’Alui sio, Faith. What the World Eats . ............................................................................................... 79 Arnosky, Jim. Wild Tracks! A Guide to Nature’s Footprints ................................................................ 79 Deedy, Carmen Agra. 14 Cows for America . ...................................................................................... 80 APPENDIX B 7 > OREGON COMMON CORE STATE STANDARDS FOR English Language Arts > & Literacy in History/Social Studies, Science, and Technical Subjects ## Sample Performance Tasks for Informational Texts .......................................................... 81 ## Grades 4 –5 Text Exemplars .................................................................................................. 82 ## Stories ............................................................................................................................. 82 Carroll, Lewis. Alice’s Adventures in Wonderland . ............................................................................. 82 Burnett, Frances Hodgson. The Secret Garden . ................................................................................. 82 Farley, Walter. The Black Stallion . ...................................................................................................... 83 Saint-Exupéry, Antoine de. The Little Prince . ..................................................................................... 83 Babbitt, Natalie. Tuck Everlasting . ..................................................................................................... 83 Singer, Isaac Bashevis. “Zlateh the Goat.” ......................................................................................... 84 Hamilton, Virginia. M. C. Higgins, the Great . ..................................................................................... 84 Erdrich, Louise. The Birchbark House . ................................................................................................ 85 Curtis, Christopher Paul. Bud, Not Buddy . ......................................................................................... 86 Lin, Grace. Where the Mountain Meets the Moon . ........................................................................... 86 ## Poetry ............................................................................................................................. 87 Blake, William. “The Echoing Green.” ................................................................................................ 87 Lazarus, Emma. “The New Colossus.” ................................................................................................ 88 Thayer, Ernest Lawrence. “Casey at the Bat.” .................................................................................... 88 Dickinson, Emily. “A Bird Came Down the Walk.” .............................................................................. 90 Sandburg, Carl. “Fog.” ........................................................................................................................ 90 Frost, Robert. “Dust of Snow.” ........................................................................................................... 91 Dahl, Roald. “Little Red Riding Hood and the Wolf.” ......................................................................... 91 Nichols, Grace. “They Were My People.” .......................................................................................... 91 Mora, Pat. “Words Free As Confetti.” ................................................................................................ 91 ## Sample Performance Tasks for Stories and Poetry ............................................................ 92 ## Informational Texts .........................................................................................................
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92 Berger, Melvin. Discovering Mars: The Amazing Story of the Red Planet . ........................................ 92 Carlisle, Madelyn Wood. Let’s Investigate Marvelously Meaningful Maps . ...................................... 93 Lauber, Patricia. Hurricanes: Earth’s Mightiest Storms . .................................................................... 93 Otfinoski, Steve. The Kid’s Guide to Money: Earning It, Saving It, Spending It, Growing It, Sharing It ........................................................................................................................................................ 93 Wulffson, Don. Toys!: Amazing Stories Behind Some Great Inventions ............................................. 94 Schleichert, Elizabeth. “Good Pet, Bad Pet.” ..................................................................................... 94 Kavash, E. Barrie. “Ancient Mound Builders.” .................................................................................... 94 Koscielniak, Bruce. About Time: A First Look at Time and Clocks . ..................................................... 94 Banting, Erinn. England the Land . ...................................................................................................... 94 Hakim, Joy. A History of US . ............................................................................................................... 95 Ruurs, Margriet. My Librarian Is a Camel: How Books Are Brought to Children Around the World . . 95 Simon, Seymour. Horses ..................................................................................................................... 96 Montgomery, Sy. Quest for the Tree Kangaroo: An Expedition to the Cloud Forest of New Guinea . 97 Simon, Seymour. Volcanoes . .............................................................................................................. 97 Nelson, Kadir. We Are the Ship: The Story of Negro League Baseball . .............................................. 98 Hall, Leslie. “Seeing Eye to Eye.” ........................................................................................................ 98 Ronan, Colin A. “Telescopes.” .......................................................................................................... 100 Buckmaster, Henrietta. “Underground Railroad.” ........................................................................... 100 ## Sample Performance Tasks for Informational Texts ........................................................ 101 APPENDIX B 8 > OREGON COMMON CORE STATE STANDARDS FOR English Language Arts > & Literacy in History/Social Studies, Science, and Technical Subjects ## Grades 6 –8 Text Exemplars ................................................................................................. 102 ## Stories ........................................................................................................................... 102 Alcott, Louisa May. Little Women . ................................................................................................... 102 Twain, Mark. The Adventures of Tom Sawyer . ................................................................................. 103 L’Engle, Madeleine. A Wrinkle in Time . ............................................................................................ 105 Cooper, Susan. The Dark Is Rising . ................................................................................................... 105 Yep, Laurence. Dragonwings . ........................................................................................................... 106 Taylor, Mildred D. Roll of Thunder, Hear My Cry . ............................................................................ 106 Hamilton, Virginia. “The People Could Fly.” .....................................................................................
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107 Paterson, Katherine. The Tale of the Mandarin Ducks ..................................................................... 107 Ci sneros, Sandra. “Eleven.” .............................................................................................................. 108 Sutcliff, Rosemary. Black Ships Before Troy: The Story of the Iliad . ................................................. 108 ## Drama ........................................................................................................................... 109 Fletcher, Louise. Sorry, Wrong Number . .......................................................................................... 109 Goodrich, Frances and Albert Hackett. The Diary of Anne Frank: A Play . ....................................... 110 ## Poetry ........................................................................................................................... 110 Longfellow, Henry Wadsworth. “Paul Revere’s Ride.” ..................................................................... 110 Whitman, Walt. “O Captain! My Captain!” ...................................................................................... 113 Carroll, Lewis. “Jabberwocky.” ......................................................................................................... 114 Navajo tradition. “Twelfth Song of Thunder.” ................................................................................. 115 Dickinson, Emily. “The Railway Train.” ............................................................................................. 115 Yeats, William Butler. “The Song of Wandering Aengus.” ............................................................... 116 Frost, Robert. “The Road Not Taken.” .............................................................................................. 116 Sandburg, Carl. “Chicago.” ............................................................................................................... 117 Hughes, Langston. “I, Too, Sing America.” ....................................................................................... 118 Neruda, Pablo. “The Book of Questions.” ........................................................................................ 118 Soto, Gary. “Oranges.” ..................................................................................................................... 118 Giovanni, Nikki. “A Poem for My Librarian, Mrs. Long.” .................................................................. 118 ## Sample Performance Tasks for Stories, Drama, and Poetry ............................................. 119 ## Informational Texts: English Language Arts .................................................................... 120 Adams, John. “Letter on Thomas Jefferson.” ................................................................................... 120 Douglass, Frederick. Narrative of the Life of Frederick Douglass an American Slave, Written by Himself . ........................................................................................................................................ 121 Churchill, Winston. “Blood, Toil, Tears and Sweat: Address to Parliament on May 13th, 1940.” ... 122 Petry, Ann. Harriet Tubman: Conductor on the Underground Railroad ........................................... 123 Steinbeck, John. Travels with Charley: In Search of America . .......................................................... 123 ## Sample Performance Tasks for Informational Texts: English Language Arts ..................... 124 ## Informational Texts: History/Social Studies .................................................................... 124 United States. Preamble and First Amendment to the United States Constitution. ....................... 124 Lord, Walter. A Night to Remember . ................................................................................................ 124 Isaacson, Phillip. A Short Walk through the Pyramids and through the World of Art . .................... 125 Murphy, Jim. The Great Fire
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451 . ........................................................................................................ 143 Olsen, Tillie. “I Stand Here Ironing.” ................................................................................................. 144 Achebe, Chinua. Things Fall Apart . .................................................................................................. 144 Lee, Harper. To Kill A Mockingbird . .................................................................................................. 145 Shaara, Michael. The Killer Angels . .................................................................................................. 146 Tan, Amy. The Joy Luck Club . ............................................................................................................ 146 Álvarez, Julia. In the Time of the Butterflies . .................................................................................... 147 Zusak, Marcus. The Book Thief . ........................................................................................................ 147 ## Drama ........................................................................................................................... 148 Sophocles. Oedipus Rex .................................................................................................................... 148 Shakespeare, William. The Tragedy of Macbeth .............................................................................. 150 Ibsen, Henrik. A Doll’s House ............................................................................................................ 152 Williams, Tennessee. The Glass Menagerie . .................................................................................... 153 Ionesco, Eugene. Rhinoceros . ........................................................................................................... 154 Fugard, Athol. “Master Harold”…and the boys . ............................................................................... 155 ## Poetry ........................................................................................................................... 156 APPENDIX B 10 > OREGON COMMON CORE STATE STANDARDS FOR English Language Arts > & Literacy in History/Social Studies, Science, and Technical Subjects Shakespeare, William. “Sonnet 73.” ................................................................................................ 156 Donne, John. “Song.” ....................................................................................................................... 156 Shelley, Percy Bysshe. “Ozymandias.” ............................................................................................. 157 Poe, Edgar Allen. “The Raven.” ........................................................................................................ 158 Dickinson, Emily. “We Grow Accustomed to the Dark.” .................................................................. 160 Houseman, A. E. “Loveliest of Trees.” .............................................................................................. 161 Johnson, James Weldon. “Lift Every Voice and Sing.” ..................................................................... 161 Cullen, Countee. “Yet Do I Marvel.” ................................................................................................. 162 Auden, Wystan Hugh. ”Musée des Beaux Arts.” ............................................................................. 162 Walker, Alice. “Women.” ................................................................................................................. 162 Baca, Jimmy Santiago. “I Am Offering This Poem to You.” .............................................................. 162 ## Sample Performance Tasks for Stories, Drama, and Poetry ............................................. 163 ## Informational Texts: English Language Arts .................................................................... 164 Henry, Patrick. “Speech to the Second Virginia Convention.” ......................................................... 164 Washington, George. “Farewell Address.” ....................................................................................... 165 Lincoln, Abraham. “Gettysburg Address.” ....................................................................................... 166 Lincoln, Abraham. “Second Inaugural Address.” ............................................................................. 167 Roosevelt , Franklin Delano. “State of the Union Address.” ............................................................. 168 Hand, Learned. “I Am an American Day Address.” .......................................................................... 169 Smith, Margaret Chase. “Remarks to the
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