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9 19 h. P(A|B) = P(AANDB) P(B) + 10 19 = 1) = 3 6, P(B|A) = P(AANDB) P(A), No = 3 9 3.1 The sample space S is the ordered pairs of two whole numbers, the first from one to three and the second from one to four (Example: (1, 4)). a. S = _____________________________ Let event A = the sum is even and event B = the first... |
1 2 e. A|B = {2}, P(A|B) = 1 3 f. B|A = {2}, P(B|A) = 1 3 g. A AND B = {2}, P(A AND B) = 1 6 h. A OR B = {1, 2, 3, 4, 6}, P(A OR B) = 5 6 i. A OR B′ = {2, 4, 5, 6}, P(A OR B′) = 2 3 j. N = {2, 3, 5}, P(N) = 1 2 k. A six-sided die does not have seven dots. P(7) = 0. Example 3.3 Table 3.1 describes the distribution of a... |
.2 | Independent and Mutually Exclusive Events Independent and mutually exclusive do not mean the same thing. Independent Events Two events are independent if the following are true: • P(A|B) = P(A) • P(B|A) = P(B) • P(A AND B) = P(A)P(B) Two events A and B are independent if the knowledge that one occurred does not af... |
reshuffle the cards and pick a third card from the 52-card deck. This time, the card is the Q of spades again. Your picks are {Q of spades, ten of clubs, Q of spades}. You have picked the Q of spades twice. You pick each card from the 52-card deck. b. Sampling without replacement: Suppose you pick three cards without ... |
7D, 6D, KH. Which of a. or b. did you sample with replacement and which did you sample without replacement? Solution 3.5 a. Without replacement; b. With replacement 3.5 You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are 13 cards in e... |
event of getting at most one tail. (At most one tail means zero or one tail.) Then A can be written as {HH, HT, TH}. The outcome HH shows zero tails. HT and TH each show one tail. • Let B = the event of getting all tails. B can be written as {TT}. B is the complement of A, so B = A′. Also, P(A) + P(B) = P(A) + P(A′) =... |
one (1) tails occur when the outcomes HH, TH, HT show up. P(F) = 3 4 b. Two faces are the same if HH or TT show up. P(G) = 2 4 c. A head on the first flip followed by a head or tail on the second flip occurs when HH or HT show up. P(H) = 2 4 d. F and G share HH so P(F AND G) is not equal to zero (0). F and G are not m... |
no.) Why or why not? Solution 3.8 No. C = {3, 5} and E = {1, 2, 3, 4}. P(C AND E) = 1 6. To be mutually exclusive, P(C AND E) must be zero. • Find P(C|A). This is a conditional probability. Recall that the event C is {3, 5} and event A is {1, 3, 5}. To find P(C|A), find the probability of C using the sample space A. Y... |
P(H) = (0.6)(0.5) = 0.3 = P(G AND H) Since G and H are independent, knowing that a person is taking a science class does not change the chance that he or she is taking a math class. If the two events had not been independent (that is, they are dependent) then knowing that a person is taking a science class would change... |
) = 0.30 and P(B AND D) = 0.20. a. Find P(B|D). b. Find P(D|B). c. Are B and D independent? d. Are B and D mutually exclusive? Example 3.11 In a box there are three red cards and five blue cards. The red cards are marked with the numbers 1, 2, and 3, and the blue cards are marked with the numbers 1, 2, 3, 4, and 5. The... |
for the home team. • 25% of the fans are wearing blue. • 20% of the fans are wearing blue and are rooting for the away team. • Of the fans rooting for the away team, 67% are wearing blue. Let A be the event that a fan is rooting for the away team. Let B be the event that a fan is wearing blue. Are the events of rootin... |
P(I AND F) = 0 because Mark will take only one route to work. What is the probability of P(I OR F)? This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 3 | PROBABILITY TOPICS 175 Example 3.13 a. Toss one fair coin (the coin has two ... |
. Yes, because P(A AND B) = 0 h. P(A AND B) = 0.P(A)P(B) = ⎛ ⎝ ⎞ ⎠ 3 12 ⎛ ⎝ 1 12 ⎞ ⎠. P(A AND B) does not equal P(A)P(B), so A and B are dependent. 3.13 A box has two balls, one white and one red. We select one ball, put it back in the box, and select a second ball (sampling with replacement). Let T be the event of get... |
= Alaska • Klaus can only afford one vacation. The probability that he chooses A is P(A) = 0.6 and the probability that he chooses B is P(B) = 0.35. • P(A AND B) = 0 because Klaus can only afford to take one vacation • Therefore, the probability that he chooses either New Zealand or Alaska is P(A OR B) = P(A) + P(B) =... |
and B mutually exclusive? This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 3 | PROBABILITY TOPICS 177 Solution 3.15 d. No, they are not because P(A and B) = 0.585. To be mutually exclusive, P(A AND B) must equal zero. 3.15 Helen ... |
≠ 0.0996 178 CHAPTER 3 | PROBABILITY TOPICS 3.16 A school has 200 seniors of whom 140 will be going to college next year. Forty will be going directly to work. The remainder are taking a gap year. Fifty of the seniors going to college play sports. Thirty of the seniors going directly to work play sports. Five of the s... |
? What is the probability that woman tests negative? Solution 3.18 a. P(B) = 0.143; P(N) = 0.85 b. Given that the woman has breast cancer, what is the probability that she tests negative? Solution 3.18 b. P(N|B) = 0.02 c. What is the probability that the woman has breast cancer AND tests negative? This content is avail... |
c. What is the probability that a woman does not develop breast cancer. Find P(B′) = 1 - P(B). d. What is the probability that a woman tests positive for breast cancer. Find P(P) = 1 - P(N). Solution 3.19 a. 0.98; b. 0.1401; c. 0.857; d. 0.15 3.19 A student goes to the library. Let events B = the student checks out a ... |
685 ⎝ 755 305 755 ⎞ ⎠ − 280 755 = 710 755 e. Find P(Person is a car phone user GIVEN person had a violation in the last year). Solution 3.20 e. 25 70 (The sample space is reduced to the number of persons who had a violation.) This content is available for free at http://textbookequity.org/introductory-statistics or at... |
�� 34 100 ⎞ ⎠ = (0.45)(0.34) = 0.153 P(F AND C) ≠ P(F)P(C), so the events F and C are not independent. c. Find the probability that a person is male given that the person prefers hiking near lakes and streams. Let M = being male, and let L = prefers hiking near lakes and streams. 1. What word tells you this is a condit... |
is 4 5. If he goes out the second door, the probability he gets caught by Alissa is 1 4 and the probability he is not caught is 3 4. The probability that Alissa catches Muddy coming out of the third door is 1 2 and the probability she does not catch Muddy is 1 2 Muddy will choose any of the three doors so the probabil... |
|Rape). e. Find P(Vehicle|2008). Solution 3.23 a. 0.0294, b. 0.1551, c. 0.7165, d. 0.2365, e. 0.2575 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 3 | PROBABILITY TOPICS 185 3.23 Table 3.10 relates the weights and heights of a ... |
is distinct. In fact, we can list each red ball as R1, R2, and R3 and each blue ball as B1, B2, B3, B4, B5, B6, B7, and B8. Then the nine RR outcomes can be written as: R1R1; R1R2; R1R3; R2R1; R2R2; R2R3; R3R1; R3R2; R3R3 The other outcomes are similar. There are a total of 11 balls in the urn. Draw two balls, one at ... |
⎞ ⎠ 8 11 = 24 121 e. Using the tree diagram, calculate P(R on 2nd draw GIVEN B on 1st draw). Solution 3.24 e. P(R on 2nd draw GIVEN B on 1st draw) = P(R on 2nd|B on 1st) = 24 88 = 3 11 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHA... |
. 188 CHAPTER 3 | PROBABILITY TOPICS Figure 3.4 Total = 56 + 24 + 24 + 6 110 = 110 110 = 1 NOTE If you draw a red on the first draw from the three red possibilities, there are two red marbles left to draw on the second draw. You do not put back or replace the first marble after you have drawn it. You draw without repla... |
(B on 2nd|R on 1st). Solution 3.25 f. Using the tree diagram, P(B on 2nd|R on 1st) = P(R|B) = 8 10. If we are using probabilities, we can label the tree in the following general way. • P(R|R) here means P(R on 2nd|R on 1st) • P(B|R) here means P(B on 2nd|R on 1st) • P(R|B) here means P(R on 2nd|B on 1st) • P(B|B) here ... |
agram A Venn diagram is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events. Example 3.27 Suppose an experiment has the outcomes 1, 2, 3,..., 12 where each outcome has an equal ch... |
. Let C = student belongs to a club and PT = student works part time. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 3 | PROBABILITY TOPICS 193 Figure 3.8 If a student is selected at random, find • • • • • the probability that t... |
f. In the Venn Diagram, describe the overlapping area using a complete sentence. In the Venn Diagram, describe the area in the rectangle but outside both the circle and the oval using a complete sentence. Solution 3.30 a. 0.51; b. 0.075; c. 0.04; d. 0.545; e. The area represents the African Americans that have type O ... |
After you record the pick, put both M&Ms back. Do this a total of 24 times, also. Use the data from Table 3.14 to calculate the empirical probability questions. Leave your answers in unreduced fractional form. Do not multiply out any fractions. Color Quantity Yellow (Y) Green (G) Blue (BL) Brown (B) Orange (O) Red (R)... |
3.14 Empirical Probabilities Discussion Questions 1. Why are the “With Replacement” and “Without Replacement” probabilities different? 2. Convert P(no yellows) to decimal format for both Theoretical “With Replacement” and for Empirical “With Replacement”. Round to four decimal places. a. Theoretical “With Replacement”... |
and B are independent if one of the following is true: 1. P(A|B) = P(A) 2. P(B|A) = P(B) 3. P(A AND B) = P(A)P(B) Mutually Exclusive Two events are mutually exclusive if the probability that they both happen at the same time is zero. If events A and B are mutually exclusive, then P(A AND B) = 0. Outcome a particular r... |
quity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 Venn Diagram the visual representation of a sample space and events in the form of circles or ovals showing their CHAPTER 3 | PROBABILITY TOPICS 199 intersections CHAPTER REVIEW 3.1 Terminology In this module we learned the basic terminology o... |
OR event, the AND event, and the complement of an event and for understanding conditional probabilities. FORMULA REVIEW 3.1 Terminology A and B are events P(S) = 1 where S is the sample space 0 ≤ P(A) ≤ 1 P(A|B) = P(AANDB) P(B) 3.2 Independent and Mutually Exclusive Events PRACTICE 3.1 Terminology If A and B are indep... |
information to answer the next four exercises. A box is filled with several party favors. It contains 12 hats, 15 noisemakers, ten finger traps, and five bags of confetti. Let H = the event of getting a hat. Let N = the event of getting a noisemaker. Let F = the event of getting a finger trap. Let C = the event of get... |
probability of drawing a club in a standard deck of 52 cards? 20. What is the probability of rolling an even number of dots with a fair, six-sided die numbered one through six? 21. What is the probability of rolling a prime number of dots with a fair, six-sided die numbered one through six? Use the following informati... |
set of all possible outcomes? 35. What is conditional probability? 36. A shelf holds 12 books. Eight are fiction and the rest are nonfiction. Each is a different book with a unique title. The fiction books are numbered one to eight. The nonfiction books are numbered one to four. Randomly select one book Let F = event ... |
L? 48. Find P(L AND C). 49. In words, what is L AND C? 50. Are L and C independent events? Show why or why not. 51. Find P(L OR C). 52. In words, what is L OR C? 53. Are L and C mutually exclusive events? Show why or why not. 3.4 Contingency Tables Use the following information to answer the next four exercises. Table ... |
30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 Whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 Whites. 59. Comple... |
what percent are more than 44 years old? j. Find P(Approve|Age < 35). 67. Explain what is wrong with the following statements. Use complete sentences. a. If there is a 60% chance of rain on Saturday and a 70% chance of rain on Sunday, then there is a 130% chance of rain over the weekend. b. The probability that a base... |
California registered voters), the approval rating was 78%. Six in ten California registered voters said that the upcoming Supreme Court’s ruling about the constitutionality of California’s Proposition 8 was either very or somewhat important to them. Out of those CA registered voters who support same-sex marriage, 75% ... |
number is assigned to a color and a range. Figure 3.13 (credit: film8ker/wikibooks) 82. a. List the sample space of the 38 possible outcomes in roulette. b. You bet on red. Find P(red). c. You bet on -1st 12- (1st Dozen). Find P(-1st 12-). d. You bet on an even number. Find P(even number). e. f. Find two mutually excl... |
E) = _____ d. P(G AND E) = _____ e. P(G OR E) = _____ f. Are G and E mutually exclusive? Justify your answer numerically. 86. Roll two fair dice. Each die has six faces. a. List the sample space. b. Let A be the event that either a three or four is rolled first, followed by an even number. Find P(A). c. Let B be the ev... |
there are at least two tails. Find P(A). c. Let B be the event that the first and second tosses land on heads. Are the events A and B mutually exclusive? Explain your answer in one to three complete sentences, including justification. 90. Consider the following scenario: Let P(C) = 0.4. Let P(D) = 0.5. Let P(C|D) = 0.... |
one of approximately 6.5 million people who entered this lottery. Let G = won green card. a. What was Renate’s chance of winning a Green Card? Write your answer as a probability statement. b. In the summer of 1994, Renate received a letter stating she was one of 110,000 finalists chosen. Once the finalists were chosen... |
hit by Hank Aaron) b. No, because P(hit by Hank Aaron|hit is a double) ≠ P(hit is a double) c. No, because P(hit is by Hank Aaron|hit is a double) ≠ P(hit by Hank Aaron) d. Yes, because P(hit is by Hank Aaron|hit is a double) = P(hit is a double) This content is available for free at http://textbookequity.org/introduct... |
(E|D). c. Find P(D OR E). d. Using an appropriate test, show whether D and E are independent. e. Using an appropriate test, show whether D and E are mutually exclusive. 3.4 Contingency Tables Use the information in the Table 3.19 to answer the next eight exercises. The table shows the political party affiliation of eac... |
310 4,650 18,780 29,760 Do not include "all others" for parts f and g. a. Fill in the column for the suicides for individuals over age 64. b. Fill in the row for all other races. c. Find the probability that a randomly selected individual was a white male. d. Find the probability that a randomly selected individual wa... |
or she has straight hair? f. g. If B is the event of a child having brown hair, find the probability of the complement of B. In words, what does the complement of B represent? 113. In a previous year, the weights of the members of the San Francisco 49ers and the Dallas Cowboys were published in the San Jose Mercury Ne... |
cookies selected were the same flavor. Find P(S). e. Let T be the event that the cookies selected were different flavors. Find P(T) by two different methods: by using the complement rule and by using the branches of the tree. Your answers should be the same with both methods. f. Let U be the event that the second cook... |
of the complement of event (H OR G). 120. Given events J and K: P(J) = 0.18; P(K) = 0.37; P(J OR K) = 0.45 a. Find P(J AND K). b. Find the probability of the complement of event (J AND K). c. Find the probability of the complement of event (J AND K). Use the following information to answer the next two exercises. Supp... |
being age 65 or over and being female mutually exclusive events? How do you know? 124. Suppose that 10,000 U.S. licensed drivers are randomly selected. a. How many would you expect to be male? b. Using the table or tree diagram, construct a contingency table of gender versus age group. c. Using the contingency table, ... |
____ Suppose a person with AIDS in Santa Clara County is randomly selected. a. Find P(Person is female). b. Find P(Person has a risk factor heterosexual contact). c. Find P(Person is female OR has a risk factor of IV drug user). d. Find P(Person is female AND has a risk factor of homosexual/bisexual). e. Find P(Person ... |
Field. “The File Poll.” Field Research Corporation. Available online at http://www.field.com/ fieldpollonline/subscribers/Rls2443.pdf (accessed May 2, 2013). Rider, David, “Ford support plummeting, poll suggests,” The Star, September 14, 2011. Available online at http://www.thestar.com/news/gta/2011/09/14/ford_support... |
A., Daniel O. Stram, Lynn R. Wilkens, Malcom C. Pike, Laurence N. Kolonel, Brien E. Henderson, and Loīc Le Marchand. “Ethnic and Racial Differences in the Smoking-Related Risk of Lung Cancer.” The New England Journal of Medicine, 2013. Available online at http://www.nejm.org/doi/full/10.1056/NEJMoa033250 (accessed May... |
F) g. P(F|L) h. P(F OR L) i. P(M AND S) j. P(F) 3 P(N) = 15 42 = 5 14 = 0.36 5 P(C) = 5 42 = 0.12 7 P(G) = 20 150 = 2 15 = 0.13 9 P(R) = 22 150 = 11 75 = 0.15 11 P(O) = 150 - 22 - 38 - 20 - 28 - 26 150 = 16 150 = 8 75 = 0.11 13 P(E) = 47 194 = 0.24 15 P(N) = 23 194 = 0.12 17 P(S) = 12 194 = 6 97 = 0.06 19 13 52 = 1 4 =... |
�� 72 130 ⎛ ⎞ ⎝ ⎠ 62 130 ⎞ ⎠ = 4, 464 16, 900 = 1, 116 4, 225 = 0.26 No, they are not independent because P(being a female musician AND learning music in school) is not equal to P(being a female musician)P(learning music in school). 58 Figure 3.15 60 35,065 100,450 62 To pick one person from the study who is Japanese A... |
or 0-00-2; or 00-2-3) = 3 38 85 a. {G1, G2, G3, G4, G5, Y1, Y2, Y3} b. c. d. 5 8 2 3 2 8 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 3 | PROBABILITY TOPICS 219 e. 6 8 f. No, because P(G AND E) does not equal 0. 87 NOTE The c... |
the money was placed in. There are several ways to justify this mathematically, but one is that the money placed in economics classes is not returned at the same overall rate; P(R|E) ≠ P(R). e. No, this study definitely does not support that notion; in fact, it suggests the opposite. The money placed in the economics ... |
103 10 67 105 10 34 107 d 109 a. b. c. d. e. f. g. 22,050 29,760 330 29,760 2,000 29,760 23,720 29,760 5,010 6,020 111 b 113 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 3 | PROBABILITY TOPICS 221 a. b. c. d. e. 26 106 33 106... |
259 0.4186 0.0695 0.514 Totals 0.0503 0.8140 0.1356 1 Table 3.28 d. P(>64 and F) = P(F) P(>64|F) = (0.486)(0.1361) = 0.0661 e. P(>64|F) is the percentage of female drivers who are 65 or older and P(>64 and F) is the percentage of drivers who are female and 65 or older. f. P(>64) = P(>64 and F) + P(>64 and M) = 0.1356 g... |
/1.16 CHAPTER 4 | DISCRETE RANDOM VARIABLES 225 4 | DISCRETE RANDOM VARIABLES Figure 4.1 You can use probability and discrete random variables to calculate the likelihood of lightening striking the ground five times during a half-hour thunderstorm. (Credit: Leszek Leszczynski) Introduction Chapter Objectives By the end... |
the x values 0, 1, 2, 3), X is a discrete random variable. Toss a coin ten times and record the number of heads. After all members of the class have completed the experiment (tossed a coin ten times and counted the number of heads), fill in Table 4.1. Let X = the number of heads in ten tosses of the coin. x Frequency o... |
. For a random sample of 50 patients, the following information was obtained. Let X = the number of times a patient rings the nurse during a 12-hour shift. For this exercise, x = 0, 1, 2, 3, 4, 5. P(x) = the probability that X takes on value x. Why is this a discrete probability distribution function (two reasons)? X P... |
tail? You might toss a fair coin ten times and record nine heads. As you learned in Section 3., probability does not describe the short-term results of an experiment. It gives information about what can be expected in the long term. To demonstrate this, Karl Pearson once tossed a fair coin 24,000 times! He recorded th... |
an expected value table. The table helps you calculate the expected value or long-term average. Add the last column x*P(x) to find the long term average or expected value: (0)(0.2) + (1)(0.5) + (2)(0.3) = 0 + 0.5 + 0.6 = 1.1. The expected value is 1.1. The men's soccer team would, on the average, expect to play soccer... |
50 (4) ⎛ ⎝ ⎞ ⎠ 4 50 = 16 50 (4 – 2.1)2 ⋅ 0.08 = 0.2888 P(x = 5) = 1 50 (5) ⎛ ⎝ ⎞ ⎠ 1 50 = 5 50 (5 – 2.1)2 ⋅ 0.02 = 0.1682 Table 4.6 You expect a newborn to wake its mother after midnight 2.1 times per week, on the average. Add the values in the third column of the table to find the expected value of X: μ = Expected Va... |
your $2 back plus $100,000). Over the long term, what is your expected profit of playing the game? To do this problem, set up an expected value table for the amount of money you can profit. Let X = the amount of money you profit. The values of x are not 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Since you are interested in your pr... |
.6 Suppose you play a game with a biased coin. You play each game by tossing the coin once. P(heads) = 2 3 and. If you toss a head, you pay $6. If you toss a tail, you win $10. If you play this game many times, P(tails) = 1 3 will you come out ahead? a. Define a random variable X. Solution 4.6 a. X = amount of profit 2... |
x) x*P(x) (x – μ)2P(x) 0 1 2 0.2 0.5 0.3 (0)(0.2) = 0 (0 – 1.1)2(0.2) = 0.242 (1)(0.5) = 0.5 (1 – 1.1)2(0.5) = 0.005 (2)(0.3) = 0.6 (2 – 1.1)2(0.3) = 0.243 Table 4.12 Add the last column in the table. 0.242 + 0.005 + 0.243 = 0.490. The standard deviation is the square root of 0.49, or σ = 0.49 = 0.7 Generally for proba... |
2 9 36 18 36 9 36 0 18 36 18 36 (0 – 1)2 ⋅ 9 36 = 9 36 (1 – 1)2 ⋅ 18 36 = 0 (1 – 1)2 ⋅ 9 36 = 9 36 Table 4.14 Calculating μ and σ. 234 CHAPTER 4 | DISCRETE RANDOM VARIABLES Add the values in the third column to find the expected value: μ = 36 36 = 1. Use this value to complete the fourth column. Add the values in the ... |
the next 48 hours in Japan was about 1.08%. As in Example 4.8, you bet that a moderate earthquake will occur in Japan during this period. If you win the bet, you win $100. If you lose the bet, you pay $10. Let X = the amount of profit from a bet. Find the mean and standard deviation of X. Some of the more common discr... |
The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X = the number of successes obtained in the n independent trials. The mean, μ, and variance, σ2, for the binomial probability distribution are μ = np and σ2 = npq. The standard deviation, σ, is then σ = npq. Any experime... |
times. Each flip is independent. What is the probability of getting more than ten heads? Let X = the number of heads in 15 flips of the fair coin. X takes on the values 0, 1, 2, 3,..., 15. Since the coin is fair, p = 0.5 and q = 0.5. The number of trials is n = 15. State the probability question mathematically. Soluti... |
�s exam on the first try. A group of 50 individuals who have taken the driver’s exam is randomly selected. Give two reasons why this is a binomial problem. Notation for the Binomial: B = Binomial Probability Distribution Function X ~ B(n, p) Read this as "X is a random variable with a binomial distribution." The parame... |
.2 The y-axis contains the probability of x, where X = the number of workers who have only a high school diploma. The number of adult workers that you expect to have a high school diploma but not pursue any further education is the mean, μ = np = (20)(0.41) = 8.2. The formula for the variance is σ2 = npq. The standard ... |
1 – 0.9443 = 0.0557 c. i. Mean = np = (100) ⎛ ⎝ ⎞ ⎠ 8 560 = 800 560 ≈ 1.4286 ⎛ ii. Standard Deviation = npq = (100) ⎝ 8 560 ⎞ ⎛ ⎝ ⎠ ⎞ ⎠ 552 560 ≈ 1.1867 4.14 According to a Gallup poll, 60% of American adults prefer saving over spending. Let X = the number of American adults out of a random sample of 50 who prefer sav... |
will develop cancer than six. 240 CHAPTER 4 | DISCRETE RANDOM VARIABLES 4.15 During the 2013 regular NBA season, DeAndre Jordan of the Los Angeles Clippers had the highest field goal completion rate in the league. DeAndre scored with 61.3% of his shots. Suppose you choose a random sample of 80 shots made by DeAndre du... |
e until you hit the bullseye. The first time you hit the bullseye is a "success" so you stop throwing the dart. It might take six tries until you hit the bullseye. You can think of the trials as failure, failure, failure, failure, failure, success, STOP. 2. In theory, the number of trials could go on forever. There mus... |
. On average, how many reports would the safety engineer expect to look at until she finds a report showing an accident caused by employee failure to follow instructions? What is the probability that the safety engineer will have to examine at least three reports until she finds a report showing an accident caused by e... |
to find a store that carries a special printer ink. You know that of the stores that carry printer ink, 10% of them carry the special ink. You randomly call each store until one has the ink you need. What are p and q? Notation for the Geometric: G = Geometric Probability Distribution Function X ~ G(p) Read this as "X ... |
�� 1 0.02 − 1 ⎞ ⎠ = 49.5 4.20 The probability of a defective steel rod is 0.01. Steel rods are selected at random. Find the probability that the first defect occurs on the ninth steel rod. Use the TI-83+ or TI-84 calculator to find the answer. Example 4.21 The lifetime risk of developing pancreatic cancer is about one ... |
The team consists of ten players. 4. Each pick is not independent, since sampling is without replacement. In the softball example, the probability of picking a woman first is 13 24. The probability of picking a man second is 11 23 if a woman was picked first. It is 10 23 if a man was picked first. The probability of t... |
DVD players. (They may be non-defective or defective.) Let X = the number of defective DVD players in the sample of 12. X takes on the values 0, 1, 2,..., 10. X may not take on the values 11 or 12. The sample size is 12, but there are only 10 defective DVD players. Write the probability statement mathematically. Solut... |
with a hypergeometric distribution." The parameters are r, b, and n; r = the size of the group of interest (first group), b = the size of the second group, n = the size of the chosen sample. Example 4.25 A school site committee is to be chosen randomly from six men and five women. If the committee consists of four mem... |
book. It might be that, on the average, there are five words spelled incorrectly in 100 pages. The interval is the 100 pages. 2. The Poisson distribution may be used to approximate the binomial if the probability of success is "small" (such as 0.01) and the number of trials is "large" (such as 1,000). You will verify ... |
You notice that a news reporter says "uh," on average, two times per broadcast. What is the probability that the news reporter says "uh" more than two times per broadcast. This is a Poisson problem because you are interested in knowing the number of times the news reporter says "uh" during a broadcast. a. What is the ... |
> 1) = 0.1734 (calculator or computer) • Press 1 – and then press 2nd DISTR. • Arrow down to poissoncdf. Press ENTER. • Enter (.75,1). • The result is P(x > 1) = 0.1734. NOTE The TI calculators use λ (lambda) for the mean. The probability that Leah receives more than one telephone call in the next 15 minutes is about ... |
the probability that a teen girl sends exactly 175 texts per day? b. What is the probability that a teen girl sends at most 150 texts per day? c. What is the standard deviation? This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 4 ... |
39 Using the Poisson distribution: • Calculate μ = np = 200(0.0102) ≈ 2.04 • P(x = 10) = poissonpdf(2.04, 10) ≈ 0.000045 We expect the approximation to be good because n is large (greater than 20) and p is small (less than 0.05). The results are close—both probabilities reported are almost 0. 4.32 On May 13, 2013, star... |
/col11562/1.16 a. b. x¯ = ________ s = ________ 3. Construct a histogram of the empirical data. CHAPTER 4 | DISCRETE RANDOM VARIABLES 253 Figure 4.6 Theoretical Distribution a. Build the theoretical PDF chart based on the distribution in the Procedure section. x P(x 10 Table 4.17 b. Calculate the following: a. μ = ____... |
distribution. • The student will demonstrate an understanding of long-term probabilities. Supplies • one “Lucky Dice” game or three regular dice Procedure Round answers to relative frequency and probability problems to four decimal places. 1. The experimental procedure is to bet on one object. Then, roll three Lucky D... |
APTER 4 | DISCRETE RANDOM VARIABLES 257 Figure 4.9 Use the Data NOTE RF = relative frequency Use the data from the Theoretical Distribution section to calculate the following answers. Round your answers to four decimal places. 1. P(x = 3) = _________________ 2. P(0 < x < 3) = _________________ 3. P(x ≥ 2) = ___________... |
�Independent” means that the result of any trial (for example, trial one) does not affect the results of the following trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV X is defined as the number of successes in n trials. The notation is: X ~ B(n, p). The mean is ... |
successes out of the total number of items chosen. Notation: X ~ H(r, b, n), where r = the number of items in the group of interest, b = the number of items in the group not of interest, and n = the number of items chosen. Mean of a Probability Distribution the long-term average of many trials of a statistical experim... |
, Y, Z,...; common notation for a specific value from the domain (set of all possible values of a variable) are lower case Latin letters x, y, and z. For example, if X is the number of children in a family, then x represents a specific integer 0, 1, 2, 3,.... Variables in statistics differ from variables in intermediat... |
4 Geometric Distribution There are three characteristics of a geometric experiment: 1. There are one or more Bernoulli trials with all failures except the last one, which is a success. 2. In theory, the number of trials could go on forever. There must be at least one trial. 3. The probability, p, of a success and the p... |
Deviation: σ = ∑ x ∈ X (x − µ)2 P(x) X = the number of successes in n independent trials n = the number of independent trials X takes on the values x = 0, 1, 2, 3,..., n p = the probability of a success for any trial q = the probability of a failure for any trial.3 Binomial Distribution The mean of X is μ = np. The st... |
... The mean μ is typically given. The variance is σ2 = μ, and the standard deviation is σ = µ. When P(μ) is used to approximate a binomial distribution, μ = np where n represents the number of independent trials and p represents the probability of success in a single trial. 4.1 Probability Distribution Function (PDF) ... |
data. 12. We know that for a probability distribution function to be discrete, it must have two characteristics. One is that the sum of the probabilities is one. What is the other characteristic? Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does ... |
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