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So trial, outcome, success, or failure. It's either gonna go either way. The result of each trial is independent from the other ones. Whether I get a six on the third trial is independent on whether I got a six on the first or the second trial. So result, let me write this trial, I'll just do a shorthand trial, results, results, independent. Independent, that's an important condition. Let's see, there are a fixed number of trials, fixed number of trials. | Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3 |
Whether I get a six on the third trial is independent on whether I got a six on the first or the second trial. So result, let me write this trial, I'll just do a shorthand trial, results, results, independent. Independent, that's an important condition. Let's see, there are a fixed number of trials, fixed number of trials. In this case, we're gonna have 12 trials. And then the last one is we have the same probability on each trial. Same probability of success. | Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3 |
Let's see, there are a fixed number of trials, fixed number of trials. In this case, we're gonna have 12 trials. And then the last one is we have the same probability on each trial. Same probability of success. Probability on each trial. So yes, indeed, this met all the conditions for being a binomial, binomial random, random variable. And this was all just a little bit of review about things that we have talked about in other videos. | Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3 |
Same probability of success. Probability on each trial. So yes, indeed, this met all the conditions for being a binomial, binomial random, random variable. And this was all just a little bit of review about things that we have talked about in other videos. But what about this thing in the salmon color, the random variable Y? So this says the number of rolls until we get a six on a fair die. So this one strikes us as a little bit different, but let's see where it is actually different. | Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3 |
And this was all just a little bit of review about things that we have talked about in other videos. But what about this thing in the salmon color, the random variable Y? So this says the number of rolls until we get a six on a fair die. So this one strikes us as a little bit different, but let's see where it is actually different. So does it meet that the trial outcomes, that there's a clear success or failure for each trial? Well, yeah, we're just gonna keep rolling. So each time we roll, it's a trial. | Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3 |
So this one strikes us as a little bit different, but let's see where it is actually different. So does it meet that the trial outcomes, that there's a clear success or failure for each trial? Well, yeah, we're just gonna keep rolling. So each time we roll, it's a trial. And success is when we get a six, failure is when we don't get a six. So the outcome of each trial can be classified as either a success or failure. So it meets, and let me put the checks right over here, it meets this first constraint. | Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3 |
So each time we roll, it's a trial. And success is when we get a six, failure is when we don't get a six. So the outcome of each trial can be classified as either a success or failure. So it meets, and let me put the checks right over here, it meets this first constraint. Are the results of each trial independent? Well, whether I get a six on the first roll or the second roll or the third roll or the fourth roll or the third roll, the probabilities shouldn't be dependent on whether I did or didn't get a six on a previous roll. So we have the independence. | Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3 |
So it meets, and let me put the checks right over here, it meets this first constraint. Are the results of each trial independent? Well, whether I get a six on the first roll or the second roll or the third roll or the fourth roll or the third roll, the probabilities shouldn't be dependent on whether I did or didn't get a six on a previous roll. So we have the independence. And we also have the same probability of success on each trial. In every case, it's a 1 6th probability that I get a six. So this stays constant. | Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3 |
So we have the independence. And we also have the same probability of success on each trial. In every case, it's a 1 6th probability that I get a six. So this stays constant. And I skipped this third condition for a reason. Because we clearly don't have a fixed number of trials. Over here, we could roll 50 times until we get a six. | Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3 |
So this stays constant. And I skipped this third condition for a reason. Because we clearly don't have a fixed number of trials. Over here, we could roll 50 times until we get a six. The probability that we'd have to roll 50 times is very low, but we might have to roll 500 times in order to get a six. In fact, think about what the minimum value of y is and what the maximum value of y is. So the minimum value that this random variable can take, I'll just call it min y, is equal to what? | Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3 |
Over here, we could roll 50 times until we get a six. The probability that we'd have to roll 50 times is very low, but we might have to roll 500 times in order to get a six. In fact, think about what the minimum value of y is and what the maximum value of y is. So the minimum value that this random variable can take, I'll just call it min y, is equal to what? Well, it's gonna take at least one roll, so that's the minimum value. But what is the maximum value for y? And I'll let you think about that. | Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3 |
So the minimum value that this random variable can take, I'll just call it min y, is equal to what? Well, it's gonna take at least one roll, so that's the minimum value. But what is the maximum value for y? And I'll let you think about that. Well, I've assumed you've thought about it if you paused the video. Well, there is no max value. You can't say, oh, it's a billion, because there's some probability that it might take a billion and one rolls. | Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3 |
And I'll let you think about that. Well, I've assumed you've thought about it if you paused the video. Well, there is no max value. You can't say, oh, it's a billion, because there's some probability that it might take a billion and one rolls. That is a very, very, very, very, very, very small probability, but there's some probability it could take a Google rolls, a Googleplex rolls, so you can imagine where this is going. So this type of random variable, where it meets a lot of the constraints of a binomial random variable, each trial has a clear success or failure outcome, the probability of success on each trial is constant, the trial results are independent of each other, but we don't have a fixed number of trials. In fact, it's a situation where we're saying, how many trials do we need to get, do we need to have until we get success? | Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3 |
You can't say, oh, it's a billion, because there's some probability that it might take a billion and one rolls. That is a very, very, very, very, very, very small probability, but there's some probability it could take a Google rolls, a Googleplex rolls, so you can imagine where this is going. So this type of random variable, where it meets a lot of the constraints of a binomial random variable, each trial has a clear success or failure outcome, the probability of success on each trial is constant, the trial results are independent of each other, but we don't have a fixed number of trials. In fact, it's a situation where we're saying, how many trials do we need to get, do we need to have until we get success? Maybe that's a general way of framing this type of random variable. How many trials until success, while the binomial random variable was, how many trials, or how many successes, I should say, how many successes in finite number of trials? So if you see this general form and it meets these conditions, you can feel good it's a binomial random variable, but if you're meeting this condition, clear success or failure outcome, independent trials, constant probability, but we're not talking about the successes in a finite number of trials, we're talking about how many trials until success, then this type of random variable is called a geometric, geometric random variable. | Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3 |
In fact, it's a situation where we're saying, how many trials do we need to get, do we need to have until we get success? Maybe that's a general way of framing this type of random variable. How many trials until success, while the binomial random variable was, how many trials, or how many successes, I should say, how many successes in finite number of trials? So if you see this general form and it meets these conditions, you can feel good it's a binomial random variable, but if you're meeting this condition, clear success or failure outcome, independent trials, constant probability, but we're not talking about the successes in a finite number of trials, we're talking about how many trials until success, then this type of random variable is called a geometric, geometric random variable. And we will see why in future videos it is called geometric because the math that involves the probabilities of various outcomes looks a lot like geometric growth or geometric sequences and series that we look at in other types of mathematics. And in case I forgot to mention, the reason why we call binomial random variables is because when you think about the probabilities of different outcomes, you have these things called binomial coefficients based on combinatorics, and those come out of things like Pascal's triangle and when you take a binomial to ever-increasing powers. So that's where those words come from. | Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3 |
And so the oldest sibling right over here, he decides, well look, I'll just put all of our names into a bowl, and then I'll just randomly pick one of our names out of the bowl each night, and then that person is going to be, so this is the bowl right over here, and I'm just gonna put four sheets of paper in there, each of them's gonna have one of their names, and then he's just going to randomly pick it out each night and then that's the person who's going to do their dishes. So they all say, well, you know, that seems like a reasonably fair thing to do, and so they start that process. So let's say that after the first three nights, that he, the oldest brother here, and let's call him Bill, let's say after three nights, Bill has not had to do the dishes. So at that point, the rest of the siblings are starting to think maybe, just maybe something fishy is happening. So what I wanna think about is, what is the probability of that happening? What's the probability of three nights in a row, Bill does not get picked? If we assume that we were randomly taking, if Bill was truly randomly taking these things out of the bowl and not cheating in some way, what's the probability that that would happen, that three nights in a row, Bill would not be picked? | Simple hypothesis testing Probability and Statistics Khan Academy.mp3 |
So at that point, the rest of the siblings are starting to think maybe, just maybe something fishy is happening. So what I wanna think about is, what is the probability of that happening? What's the probability of three nights in a row, Bill does not get picked? If we assume that we were randomly taking, if Bill was truly randomly taking these things out of the bowl and not cheating in some way, what's the probability that that would happen, that three nights in a row, Bill would not be picked? I encourage you to pause the video and think about that. Well, let's think about the probability that Bill's not picked on a given night. If it's truly random, so we're going to assume, we're going to assume that Bill's not cheating. | Simple hypothesis testing Probability and Statistics Khan Academy.mp3 |
If we assume that we were randomly taking, if Bill was truly randomly taking these things out of the bowl and not cheating in some way, what's the probability that that would happen, that three nights in a row, Bill would not be picked? I encourage you to pause the video and think about that. Well, let's think about the probability that Bill's not picked on a given night. If it's truly random, so we're going to assume, we're going to assume that Bill's not cheating. So assume, assume truly random, truly random and that each of the sheets of paper have a one in four chance of being picked. What's the probability that Bill does not get picked? Well, there's, so let me, the probability that, I guess I'm gonna write this, Bill not picked on a night, on a night, well, there's four equally likely outcomes and three of them result in Bill not getting picked. | Simple hypothesis testing Probability and Statistics Khan Academy.mp3 |
If it's truly random, so we're going to assume, we're going to assume that Bill's not cheating. So assume, assume truly random, truly random and that each of the sheets of paper have a one in four chance of being picked. What's the probability that Bill does not get picked? Well, there's, so let me, the probability that, I guess I'm gonna write this, Bill not picked on a night, on a night, well, there's four equally likely outcomes and three of them result in Bill not getting picked. So there's a 3 4th probability that Bill is not picked on a given night. Well, what's the probability that Bill's not picked three nights in a row? Let me write that down. | Simple hypothesis testing Probability and Statistics Khan Academy.mp3 |
Well, there's, so let me, the probability that, I guess I'm gonna write this, Bill not picked on a night, on a night, well, there's four equally likely outcomes and three of them result in Bill not getting picked. So there's a 3 4th probability that Bill is not picked on a given night. Well, what's the probability that Bill's not picked three nights in a row? Let me write that down. So the probability Bill not picked three nights in a row, well, that's the probability that he's not picked on the first night times the probability that he's not picked on the second night times the probability that he's not picked on the third night. So that's going to be three to the third power or three times three times three. So that's 27 over four to the third power. | Simple hypothesis testing Probability and Statistics Khan Academy.mp3 |
Let me write that down. So the probability Bill not picked three nights in a row, well, that's the probability that he's not picked on the first night times the probability that he's not picked on the second night times the probability that he's not picked on the third night. So that's going to be three to the third power or three times three times three. So that's 27 over four to the third power. Four times four times four is 64. And if we want to express that as a decimal, so that is 27, let me get my calculator out, that is 27 divided by 64 is equal to, and I'll just round to the nearest hundredth right here, 0.42. So that is equal to 0.42. | Simple hypothesis testing Probability and Statistics Khan Academy.mp3 |
So that's 27 over four to the third power. Four times four times four is 64. And if we want to express that as a decimal, so that is 27, let me get my calculator out, that is 27 divided by 64 is equal to, and I'll just round to the nearest hundredth right here, 0.42. So that is equal to 0.42. And so this doesn't seem that unlikely. It's a little less likely than kind of even odds, but you wouldn't question someone's credibility if there's a 42%, roughly a 42% chance that three nights in a row Bill would not be picked. So this seems like if you're assuming truly random that it's a reasonable, your hypothesis that it's truly random, there's a good chance that you're right. | Simple hypothesis testing Probability and Statistics Khan Academy.mp3 |
So that is equal to 0.42. And so this doesn't seem that unlikely. It's a little less likely than kind of even odds, but you wouldn't question someone's credibility if there's a 42%, roughly a 42% chance that three nights in a row Bill would not be picked. So this seems like if you're assuming truly random that it's a reasonable, your hypothesis that it's truly random, there's a good chance that you're right. There's a 42% chance you would have the outcome you saw if your assumption is true. But let's say you keep doing this and you trust your older brother. You know, why would he want to cheat out his younger siblings? | Simple hypothesis testing Probability and Statistics Khan Academy.mp3 |
So this seems like if you're assuming truly random that it's a reasonable, your hypothesis that it's truly random, there's a good chance that you're right. There's a 42% chance you would have the outcome you saw if your assumption is true. But let's say you keep doing this and you trust your older brother. You know, why would he want to cheat out his younger siblings? But let's say that Bill's not picked 12 nights in a row. So then everyone's starting to get a little bit, everyone's starting to get a little bit suspicious with Bill right over here. And so they say, well, you know, well, we're gonna give him the benefit of the doubt. | Simple hypothesis testing Probability and Statistics Khan Academy.mp3 |
You know, why would he want to cheat out his younger siblings? But let's say that Bill's not picked 12 nights in a row. So then everyone's starting to get a little bit, everyone's starting to get a little bit suspicious with Bill right over here. And so they say, well, you know, well, we're gonna give him the benefit of the doubt. Assuming that he's being completely honest, that this is a completely random process, what is the probability that he would not be picked 12 nights in a row? Well, just write that down. So the probability Bill, and it's really the same stuff that I just wrote up here. | Simple hypothesis testing Probability and Statistics Khan Academy.mp3 |
And so they say, well, you know, well, we're gonna give him the benefit of the doubt. Assuming that he's being completely honest, that this is a completely random process, what is the probability that he would not be picked 12 nights in a row? Well, just write that down. So the probability Bill, and it's really the same stuff that I just wrote up here. I'll just say Bill not picked 12 nights in a row. Well, that's going to be three, you're gonna take 12 3 4ths and multiply them together. It's going to be 3 4ths to the 12th power. | Simple hypothesis testing Probability and Statistics Khan Academy.mp3 |
So the probability Bill, and it's really the same stuff that I just wrote up here. I'll just say Bill not picked 12 nights in a row. Well, that's going to be three, you're gonna take 12 3 4ths and multiply them together. It's going to be 3 4ths to the 12th power. And what is this going to be equal to? Well, let's see, if you take, well, 3 4ths is, I'll just write three divided by three, divided by four, which is gonna be 0.75, to the 12th power. Now, this is a much smaller, this is now, if we actually, this is going to be 0.3, I guess we could go to one more decimal place, 0.32, or we could say, so this is 0.032, I should say, which is equal to, so this is approximately equal to, let me write that, which is equal to 3.2%. | Simple hypothesis testing Probability and Statistics Khan Academy.mp3 |
It's going to be 3 4ths to the 12th power. And what is this going to be equal to? Well, let's see, if you take, well, 3 4ths is, I'll just write three divided by three, divided by four, which is gonna be 0.75, to the 12th power. Now, this is a much smaller, this is now, if we actually, this is going to be 0.3, I guess we could go to one more decimal place, 0.32, or we could say, so this is 0.032, I should say, which is equal to, so this is approximately equal to, let me write that, which is equal to 3.2%. So now you have every right to start thinking that something is getting fishy. You could say, well, look, if there was, and this is what statisticians actually do, they often just define a threshold, hey, you know, if the probability of this happening purely by chance is more than 5%, then I'll say, well, maybe it was happening by chance. But if the probability of this happening purely by chance was, you know, and this is the threshold that statisticians often use is 5%, but that's somewhat arbitrarily defined, but this is a fairly low probability that it would happen fairly by chance. | Simple hypothesis testing Probability and Statistics Khan Academy.mp3 |
Now, this is a much smaller, this is now, if we actually, this is going to be 0.3, I guess we could go to one more decimal place, 0.32, or we could say, so this is 0.032, I should say, which is equal to, so this is approximately equal to, let me write that, which is equal to 3.2%. So now you have every right to start thinking that something is getting fishy. You could say, well, look, if there was, and this is what statisticians actually do, they often just define a threshold, hey, you know, if the probability of this happening purely by chance is more than 5%, then I'll say, well, maybe it was happening by chance. But if the probability of this happening purely by chance was, you know, and this is the threshold that statisticians often use is 5%, but that's somewhat arbitrarily defined, but this is a fairly low probability that it would happen fairly by chance. So you might be tempted to reject the hypothesis, to reject the hypothesis that it was truly random, that Bill is cheating in some way. And you could imagine, if it wasn't 12 in a row, if it was 20 in a row, then this probability becomes really, really, really, really, really small. And so your hypothesis that it's truly random starts to really come into doubt. | Simple hypothesis testing Probability and Statistics Khan Academy.mp3 |
So it says here I have a 0.35 probability of making a free throw. What is the probability of making four out of seven free throws? Well, this is a classic binomial random variable question. If we said the binomial random variable X is equal to number of made free throws from seven, I could say seven trials or seven shots, seven trials with the probability of success is equal to 0.35 for each free throw. So really this question amounts to what is the probability that my binomial random variable X is equal to four? Now what we're going to see is we can use a function on our TI-84 named binomec, or binomepdf I should say, binomepdf, which is short for binomial probability distribution function. And what you're going to want to do here is use three arguments. | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
If we said the binomial random variable X is equal to number of made free throws from seven, I could say seven trials or seven shots, seven trials with the probability of success is equal to 0.35 for each free throw. So really this question amounts to what is the probability that my binomial random variable X is equal to four? Now what we're going to see is we can use a function on our TI-84 named binomec, or binomepdf I should say, binomepdf, which is short for binomial probability distribution function. And what you're going to want to do here is use three arguments. So the first one is the number of trials. So in this case it is seven. And if you're doing it on the free response section of the AP test, you should make it very clear that that right over there is your N. The graders will actually look for that to make sure that you're not just guessing what goes where. | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
And what you're going to want to do here is use three arguments. So the first one is the number of trials. So in this case it is seven. And if you're doing it on the free response section of the AP test, you should make it very clear that that right over there is your N. The graders will actually look for that to make sure that you're not just guessing what goes where. So you would say that is my N and then you would say your probability, 0.35. And once again, if you're taking the test, you should mark that. That is your P. And then last but not least, what is the probability that a binomial variable, when you're taking seven trials with a probability of success of each of them being 0.35, that you have exactly four successes? | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
And if you're doing it on the free response section of the AP test, you should make it very clear that that right over there is your N. The graders will actually look for that to make sure that you're not just guessing what goes where. So you would say that is my N and then you would say your probability, 0.35. And once again, if you're taking the test, you should mark that. That is your P. And then last but not least, what is the probability that a binomial variable, when you're taking seven trials with a probability of success of each of them being 0.35, that you have exactly four successes? So now let's get our calculator out and actually do that. All right, so now we have our graphing calculator out. So there's a couple of ways to input this. | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
That is your P. And then last but not least, what is the probability that a binomial variable, when you're taking seven trials with a probability of success of each of them being 0.35, that you have exactly four successes? So now let's get our calculator out and actually do that. All right, so now we have our graphing calculator out. So there's a couple of ways to input this. You could just type it in directly. That could take time. You could do second and this little blue distribution here. | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
So there's a couple of ways to input this. You could just type it in directly. That could take time. You could do second and this little blue distribution here. So there you have it. In order to get to the function, you could either scroll down or you could scroll up to get to the bottom of the list and you see it right over here, binomPDF. You could do alpha A to go there really fast or you could just scroll up here, click Enter. | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
You could do second and this little blue distribution here. So there you have it. In order to get to the function, you could either scroll down or you could scroll up to get to the bottom of the list and you see it right over here, binomPDF. You could do alpha A to go there really fast or you could just scroll up here, click Enter. And then you have the number of trials that you wanna deal with. Well, we're gonna take seven trials. The probability of success in each trial is 0.35. | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
You could do alpha A to go there really fast or you could just scroll up here, click Enter. And then you have the number of trials that you wanna deal with. Well, we're gonna take seven trials. The probability of success in each trial is 0.35. And then my X value, well, I wanna find the probability that my binomial random variable is equal to four, four successes out of the trials. And now let me go to Paste and this is actually going to type in exactly what we had before. Notice this is the exact same thing. | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
The probability of success in each trial is 0.35. And then my X value, well, I wanna find the probability that my binomial random variable is equal to four, four successes out of the trials. And now let me go to Paste and this is actually going to type in exactly what we had before. Notice this is the exact same thing. So I have seven trials, P is equal to 0.35 and I wanna know the probability of having exactly four successes. And then I just click Enter and I get, there you go, 0.14. So this is equal to approximately 0.14. | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
Notice this is the exact same thing. So I have seven trials, P is equal to 0.35 and I wanna know the probability of having exactly four successes. And then I just click Enter and I get, there you go, 0.14. So this is equal to approximately 0.14. Now based on the same binomial random variable, if we're then asked what is the probability of making less than five free throws? So we could say this is the probability that X is less than five or we could say this is the probability that X is less than or equal to four. And the reason why I write it this way is because using it this way, you can now use the binomial cumulative distribution function on my calculator. | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
So this is equal to approximately 0.14. Now based on the same binomial random variable, if we're then asked what is the probability of making less than five free throws? So we could say this is the probability that X is less than five or we could say this is the probability that X is less than or equal to four. And the reason why I write it this way is because using it this way, you can now use the binomial cumulative distribution function on my calculator. So if I just type in binom, and once again, I'm gonna take seven, or binom CDF, I should say, cumulative distribution function, and I'm gonna take seven trials and the probability of success in each trial is 0.35. And now when I type in four here, it doesn't mean what is the probability that I make exactly four free throws. It is the probability that I make zero, one, two, three, or four free throws. | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
And the reason why I write it this way is because using it this way, you can now use the binomial cumulative distribution function on my calculator. So if I just type in binom, and once again, I'm gonna take seven, or binom CDF, I should say, cumulative distribution function, and I'm gonna take seven trials and the probability of success in each trial is 0.35. And now when I type in four here, it doesn't mean what is the probability that I make exactly four free throws. It is the probability that I make zero, one, two, three, or four free throws. So all of the possible outcomes of my binomial random variable up to and including this value right over here. So let me get that, let me get my calculator back. So once again, I can go to second, distribution. | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
It is the probability that I make zero, one, two, three, or four free throws. So all of the possible outcomes of my binomial random variable up to and including this value right over here. So let me get that, let me get my calculator back. So once again, I can go to second, distribution. I'll scroll up to go to the bottom of the list and here you see it, binomial cumulative distribution function. So let me go there, click enter. And once again, seven trials. | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
So once again, I can go to second, distribution. I'll scroll up to go to the bottom of the list and here you see it, binomial cumulative distribution function. So let me go there, click enter. And once again, seven trials. My P is 0.35. And my X value is four, but now this is gonna give me the probability that my binomial random variable equals four. This is going to give me the probability that I get any value up to and including four. | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
And once again, seven trials. My P is 0.35. And my X value is four, but now this is gonna give me the probability that my binomial random variable equals four. This is going to give me the probability that I get any value up to and including four. So this should be a higher probability. And there you have it. It is approximately 0.94. | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
This is going to give me the probability that I get any value up to and including four. So this should be a higher probability. And there you have it. It is approximately 0.94. So this is approximately 0.94. So hopefully you found that helpful. These calculators can be very useful, especially on something like an AP Stats exam. | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
So let's see what's going on here. The horizontal axis here, they say years since 1965. So this point right over here, this is zero years since 1965, so this really represents 1965. And we see it looks like around, let's see if I were to eyeball it, it looks like it's around 42% of Americans, just looking at this graph, I know that's not an exact number, roughly 41 or 42% of Americans smoked in 1965 based on this graph. And then five years later, this would be 1970, 10 years later, that would be 1975. And they don't sample the data, or we don't have data from every given year. This is just from some of the years that we happen to have. | Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3 |
And we see it looks like around, let's see if I were to eyeball it, it looks like it's around 42% of Americans, just looking at this graph, I know that's not an exact number, roughly 41 or 42% of Americans smoked in 1965 based on this graph. And then five years later, this would be 1970, 10 years later, that would be 1975. And they don't sample the data, or we don't have data from every given year. This is just from some of the years that we happen to have. But what is clear, it looks like we have a negative linear relationship right over here, that it would not be difficult to fit a line, so let me try to do that. So I'm just gonna eyeball it and try to fit a line to this data. So our line might look something like that. | Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3 |
This is just from some of the years that we happen to have. But what is clear, it looks like we have a negative linear relationship right over here, that it would not be difficult to fit a line, so let me try to do that. So I'm just gonna eyeball it and try to fit a line to this data. So our line might look something like that. So it looks like a pretty strong negative linear relationship. When I say it's a negative linear relationship, we see that as time increases, the percentage of smokers in the US is decreasing. So that's what makes it a negative relationship. | Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3 |
So our line might look something like that. So it looks like a pretty strong negative linear relationship. When I say it's a negative linear relationship, we see that as time increases, the percentage of smokers in the US is decreasing. So that's what makes it a negative relationship. Now what are they asking? They want us to estimate the percentage of American adults who smoked in 1945. Well 1945 would be to the left of zero. | Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3 |
So that's what makes it a negative relationship. Now what are they asking? They want us to estimate the percentage of American adults who smoked in 1945. Well 1945 would be to the left of zero. So we could even think of it as if 1945 is 20 years before 1965. So let me see if I can draw that. So 20 years before 1965. | Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3 |
Well 1945 would be to the left of zero. So we could even think of it as if 1945 is 20 years before 1965. So let me see if I can draw that. So 20 years before 1965. Let's see, this would be five years before 1965, 10 years, 15 years, 20 years before 1965. So I could even put that as negative 20 right over here. Negative 20 years since 1965, you could view as 20 years before 1965, so that would represent 1945 right over there. | Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3 |
So 20 years before 1965. Let's see, this would be five years before 1965, 10 years, 15 years, 20 years before 1965. So I could even put that as negative 20 right over here. Negative 20 years since 1965, you could view as 20 years before 1965, so that would represent 1945 right over there. And one thing that we could do is very roughly just try to extend this negative linear relationship backwards, and they allow us to do that by saying assuming the trend shown in the data has been consistent. So the trend has been consistent. This line represents the trend. | Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3 |
Negative 20 years since 1965, you could view as 20 years before 1965, so that would represent 1945 right over there. And one thing that we could do is very roughly just try to extend this negative linear relationship backwards, and they allow us to do that by saying assuming the trend shown in the data has been consistent. So the trend has been consistent. This line represents the trend. So let's just keep going backwards. Keep going backwards at the same rate. So something like that. | Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3 |
This line represents the trend. So let's just keep going backwards. Keep going backwards at the same rate. So something like that. I wanna make sure that it looks at the same, looks like it's the same rate right over here. And you could just try to eyeball it. You could say well see, 20 years ago, 1945, if I were to extend that line backwards, it looks like there were about 52% of the population was smoking. | Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3 |
So something like that. I wanna make sure that it looks at the same, looks like it's the same rate right over here. And you could just try to eyeball it. You could say well see, 20 years ago, 1945, if I were to extend that line backwards, it looks like there were about 52% of the population was smoking. This seems like we're about 52% right over here. Another way to think about it would be to actually try to calculate the rate of decline. And let's say we do it over every 20 years because that'll be useful because we're going 20 years back. | Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3 |
You could say well see, 20 years ago, 1945, if I were to extend that line backwards, it looks like there were about 52% of the population was smoking. This seems like we're about 52% right over here. Another way to think about it would be to actually try to calculate the rate of decline. And let's say we do it over every 20 years because that'll be useful because we're going 20 years back. So if we go 20 years from this point, so this is 1965, you go 20 years in the future. So that is 10 years and then that is 20 years. So my change in the horizontal is 20 years. | Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3 |
And let's say we do it over every 20 years because that'll be useful because we're going 20 years back. So if we go 20 years from this point, so this is 1965, you go 20 years in the future. So that is 10 years and then that is 20 years. So my change in the horizontal is 20 years. What's the change in the vertical? Well it looks like we have a decrease of a little bit more than 10%. Looks like it's 11 or 12% decrease. | Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3 |
So my change in the horizontal is 20 years. What's the change in the vertical? Well it looks like we have a decrease of a little bit more than 10%. Looks like it's 11 or 12% decrease. So I'll just say minus 11% right there. And let's see if that's consistent. If we were to go another 20 years. | Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3 |
Looks like it's 11 or 12% decrease. So I'll just say minus 11% right there. And let's see if that's consistent. If we were to go another 20 years. So if we go another 20 years, it looks like once again, we've gone down by about 10%. So that looks like roughly 10%. If we're following the line, it should actually be the same number. | Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3 |
If we were to go another 20 years. So if we go another 20 years, it looks like once again, we've gone down by about 10%. So that looks like roughly 10%. If we're following the line, it should actually be the same number. So let me write it this way. It's approximately down 10%. So that little squiggly line, I'm just saying approximately negative 10% every 20 years. | Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3 |
If we're following the line, it should actually be the same number. So let me write it this way. It's approximately down 10%. So that little squiggly line, I'm just saying approximately negative 10% every 20 years. Negative 10% every 20 years. So if you go back 20 years, you should increase your percentage by 20%. So this should go up by, or you should increase your percentage by 10% I should say. | Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3 |
So that little squiggly line, I'm just saying approximately negative 10% every 20 years. Negative 10% every 20 years. So if you go back 20 years, you should increase your percentage by 20%. So this should go up by, or you should increase your percentage by 10% I should say. So if we started at 41 or 42, once again, this is what we saw when we just eyeballed it. You should get to 51 or 52%. So my estimate of the percentage of American adults who smoked in 1945 would be 51 or 52%. | Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3 |
This is from ck12.org's open source flexbook, their AP Statistics flexbook. And I've taken the problems from their normal distribution chapter, so you could go to their site and actually look up these same problems. So this first problem, which of the following data sets is most likely to be normally distributed? For the other choices, explain why you believe they would not follow a normal distribution. So let's see, choice A. So this is really, my beliefs come into play. So this is unusual in the math context. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
For the other choices, explain why you believe they would not follow a normal distribution. So let's see, choice A. So this is really, my beliefs come into play. So this is unusual in the math context. It's more of a, what do I think? It's kind of an essay question. So let's see what they have here. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
So this is unusual in the math context. It's more of a, what do I think? It's kind of an essay question. So let's see what they have here. A, the hand span, measured from the tip of the thumb to the tip of the extended fifth finger. So I think they're talking about, let me see if I can draw a hand. So if that's the index finger, and then you've got the middle finger, and then you've got your ring finger, and then you've got your pinky, and the hand will look something like that. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
So let's see what they have here. A, the hand span, measured from the tip of the thumb to the tip of the extended fifth finger. So I think they're talking about, let me see if I can draw a hand. So if that's the index finger, and then you've got the middle finger, and then you've got your ring finger, and then you've got your pinky, and the hand will look something like that. I think they're talking about this distance, from the tip of the thumb to the tip of the extended fifth finger, which is a fancy way of saying the pinky, I think. They're talking about that distance right there. And they're saying, if I were to measure it of a random sample of high school seniors, what would it look like? | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
So if that's the index finger, and then you've got the middle finger, and then you've got your ring finger, and then you've got your pinky, and the hand will look something like that. I think they're talking about this distance, from the tip of the thumb to the tip of the extended fifth finger, which is a fancy way of saying the pinky, I think. They're talking about that distance right there. And they're saying, if I were to measure it of a random sample of high school seniors, what would it look like? Well, you know, how far this is, this is a combination of genetics and environmental factors. Maybe how much milk you drank, or how much you hung from your pinky from a bar while you were growing up. So I would think that it is a sum of a huge number of random processes. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
And they're saying, if I were to measure it of a random sample of high school seniors, what would it look like? Well, you know, how far this is, this is a combination of genetics and environmental factors. Maybe how much milk you drank, or how much you hung from your pinky from a bar while you were growing up. So I would think that it is a sum of a huge number of random processes. So I would guess that it is roughly normally distributed. If I look at my own hand, and my hand I don't think has grown much since I was a high school senior. It looks like, I don't know, it looks like roughly 9 inches or so. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
So I would think that it is a sum of a huge number of random processes. So I would guess that it is roughly normally distributed. If I look at my own hand, and my hand I don't think has grown much since I was a high school senior. It looks like, I don't know, it looks like roughly 9 inches or so. I play guitar, maybe that helped me stretch my hand. But it's really an essay question, so I just have to say what I feel. So I would guess that the distribution would look something like this. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
It looks like, I don't know, it looks like roughly 9 inches or so. I play guitar, maybe that helped me stretch my hand. But it's really an essay question, so I just have to say what I feel. So I would guess that the distribution would look something like this. I don't know, I've never done this, but maybe it has a mean of 8 inches or 9 inches, and it's distributed something like this. It's distributed something like this. So maybe it probably does look like a normal distribution. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
So I would guess that the distribution would look something like this. I don't know, I've never done this, but maybe it has a mean of 8 inches or 9 inches, and it's distributed something like this. It's distributed something like this. So maybe it probably does look like a normal distribution. But it probably won't be a perfect, in fact, I can guarantee you it won't be a perfect normal distribution. Because one, no one can have negative length of that span. This distance can never be negative. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
So maybe it probably does look like a normal distribution. But it probably won't be a perfect, in fact, I can guarantee you it won't be a perfect normal distribution. Because one, no one can have negative length of that span. This distance can never be negative. So they're going to have a, I guess, you can have no hand, so that would maybe be counted as 0. But the distribution wouldn't go into the negative domain, so it wouldn't be a perfect normal distribution on the left-hand side. It would really just end here at 0. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
This distance can never be negative. So they're going to have a, I guess, you can have no hand, so that would maybe be counted as 0. But the distribution wouldn't go into the negative domain, so it wouldn't be a perfect normal distribution on the left-hand side. It would really just end here at 0. And even on the right-hand side, there are some physically impossible hand lengths. No one can have a hand that's larger than the height of Earth's atmosphere or an astronomical unit. You would start touching the sun. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
It would really just end here at 0. And even on the right-hand side, there are some physically impossible hand lengths. No one can have a hand that's larger than the height of Earth's atmosphere or an astronomical unit. You would start touching the sun. There's some point at which it is physically impossible to get to. And in a true normal distribution, if I were to flip a bunch of coins, there's some very, very small probability that I could get a million heads in a row. It's almost 0, but there's some probability. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
You would start touching the sun. There's some point at which it is physically impossible to get to. And in a true normal distribution, if I were to flip a bunch of coins, there's some very, very small probability that I could get a million heads in a row. It's almost 0, but there's some probability. But in the case of hand spans, there's no way out here the probability of a human being who happens to be a high school senior having a, I don't know, one mile length hand span, that's 0. So it's not going to be a perfect normal distribution at the outliers or as we get further and further away from the mean. But I think it'll be a pretty good, in our everyday world, as good as we're going to get approximation, the normal distribution is going to be a pretty good approximation for the distribution that we see. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
It's almost 0, but there's some probability. But in the case of hand spans, there's no way out here the probability of a human being who happens to be a high school senior having a, I don't know, one mile length hand span, that's 0. So it's not going to be a perfect normal distribution at the outliers or as we get further and further away from the mean. But I think it'll be a pretty good, in our everyday world, as good as we're going to get approximation, the normal distribution is going to be a pretty good approximation for the distribution that we see. I guess one thing that, you know, this is high school seniors, when I did this, it was kind of from my point of view as a guy. And I would argue that high school seniors, guys probably have larger hands than women. So it's possible that you actually have a bimodal distribution. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
But I think it'll be a pretty good, in our everyday world, as good as we're going to get approximation, the normal distribution is going to be a pretty good approximation for the distribution that we see. I guess one thing that, you know, this is high school seniors, when I did this, it was kind of from my point of view as a guy. And I would argue that high school seniors, guys probably have larger hands than women. So it's possible that you actually have a bimodal distribution. So instead of having it like this, it's possible that the distribution looks like this. That you have one peak for guys, maybe at 8 inches, and then maybe another peak for women at, I don't know, 7 inches, and then the distribution falls off like that. So it's also possible it could be bimodal. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
So it's possible that you actually have a bimodal distribution. So instead of having it like this, it's possible that the distribution looks like this. That you have one peak for guys, maybe at 8 inches, and then maybe another peak for women at, I don't know, 7 inches, and then the distribution falls off like that. So it's also possible it could be bimodal. But in general, a normal distribution is going to be a pretty good approximation for part A of this problem. Let's see what part B, what they're asking us to describe. The annual salaries of all employees of a large shipping company. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
So it's also possible it could be bimodal. But in general, a normal distribution is going to be a pretty good approximation for part A of this problem. Let's see what part B, what they're asking us to describe. The annual salaries of all employees of a large shipping company. So if we're talking about annual salaries, you know, we have minimum wage laws, whatnot. So I would guess that any corporation, if we're talking about full-time workers at least, there's going to be some minimum salary that people have. And probably a lot of people will have that minimum salary because it'll be probably the most labor-intensive jobs. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
The annual salaries of all employees of a large shipping company. So if we're talking about annual salaries, you know, we have minimum wage laws, whatnot. So I would guess that any corporation, if we're talking about full-time workers at least, there's going to be some minimum salary that people have. And probably a lot of people will have that minimum salary because it'll be probably the most labor-intensive jobs. Most people are down there at the low end of the pay scale. And then you have your different middle-level managers and whatnot. And then you probably have this big gap, and then you probably have your true executives, maybe your CEO or whatnot, with, you know, if this mean right here is maybe $40,000 a year, and this is probably $80,000 where some of the mid-level managers lie. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
And probably a lot of people will have that minimum salary because it'll be probably the most labor-intensive jobs. Most people are down there at the low end of the pay scale. And then you have your different middle-level managers and whatnot. And then you probably have this big gap, and then you probably have your true executives, maybe your CEO or whatnot, with, you know, if this mean right here is maybe $40,000 a year, and this is probably $80,000 where some of the mid-level managers lie. But this out here, this will probably be, you know, actually if you were to draw a real, the way I've scaled it right now, this would be $80,000, this would be about $200,000, which is actually a reasonable salary for a CEO. But the reality is that this actually might get pushed way out from there. It might look something like that. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
And then you probably have this big gap, and then you probably have your true executives, maybe your CEO or whatnot, with, you know, if this mean right here is maybe $40,000 a year, and this is probably $80,000 where some of the mid-level managers lie. But this out here, this will probably be, you know, actually if you were to draw a real, the way I've scaled it right now, this would be $80,000, this would be about $200,000, which is actually a reasonable salary for a CEO. But the reality is that this actually might get pushed way out from there. It might look something like that. It might be way off the charts. You know, let's say the CEO made $5 million in a year because he cashed in a bunch of options or something. So it would be way over here, and maybe it's the CEO and a couple of other people, the CFO or the founders. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
It might look something like that. It might be way off the charts. You know, let's say the CEO made $5 million in a year because he cashed in a bunch of options or something. So it would be way over here, and maybe it's the CEO and a couple of other people, the CFO or the founders. So my guess is it definitely wouldn't be a normal distribution, and it definitely would have a second, it would be bimodal. You would have another peak over here for senior management up at the, unless we're, well, they're not saying, you know, if we're maybe in Europe, this would probably be closer to the left. But it won't be a perfect normal distribution, and you're not going to have any values below a certain threshold, below that kind of minimum wage level. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
So it would be way over here, and maybe it's the CEO and a couple of other people, the CFO or the founders. So my guess is it definitely wouldn't be a normal distribution, and it definitely would have a second, it would be bimodal. You would have another peak over here for senior management up at the, unless we're, well, they're not saying, you know, if we're maybe in Europe, this would probably be closer to the left. But it won't be a perfect normal distribution, and you're not going to have any values below a certain threshold, below that kind of minimum wage level. So I would call this, when you have a tail that goes more to the right than to the left, call this a right-skewed distribution. And since it has two humps right here, one there and one there, we could also say it's bimodal. I mean, it depends on what kind of company this is, but that would be my guess of a lot of large shipping companies' salaries. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
But it won't be a perfect normal distribution, and you're not going to have any values below a certain threshold, below that kind of minimum wage level. So I would call this, when you have a tail that goes more to the right than to the left, call this a right-skewed distribution. And since it has two humps right here, one there and one there, we could also say it's bimodal. I mean, it depends on what kind of company this is, but that would be my guess of a lot of large shipping companies' salaries. Let's look at choice C, or problem part C. The annual salaries of a random sample of 50 CEOs of major companies, 25 women and 25 men. The fact that they wrote this here, I think they maybe are implying that maybe men and women, you know, the gender gap has not been closed fully, and there is some discrepancy. So if it was just purely 50 CEOs of major companies, I would say it's probably close to a normal distribution. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
I mean, it depends on what kind of company this is, but that would be my guess of a lot of large shipping companies' salaries. Let's look at choice C, or problem part C. The annual salaries of a random sample of 50 CEOs of major companies, 25 women and 25 men. The fact that they wrote this here, I think they maybe are implying that maybe men and women, you know, the gender gap has not been closed fully, and there is some discrepancy. So if it was just purely 50 CEOs of major companies, I would say it's probably close to a normal distribution. It's probably something like, well, you know, once again, there's going to be some level below which no CEO is willing to work for, although I've heard of some cases where they work for free, but they're really getting paid in other ways. If you include all of those things, there's probably some base salary that all CEOs make at least that much, and then it goes up to some value, you know, the highest probability value, and then it probably has a long tail to the right. And this is if there were no gender gap. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
So if it was just purely 50 CEOs of major companies, I would say it's probably close to a normal distribution. It's probably something like, well, you know, once again, there's going to be some level below which no CEO is willing to work for, although I've heard of some cases where they work for free, but they're really getting paid in other ways. If you include all of those things, there's probably some base salary that all CEOs make at least that much, and then it goes up to some value, you know, the highest probability value, and then it probably has a long tail to the right. And this is if there were no gender gap. So this would just be a purely right-skewed distribution, where you have a long tail to the right. Now if you assume that there's some gender gap, then you might have two humps here, which would be a bimodal distribution. So if you assume there's some gender gap, this is part c right here, then maybe there's one hump for women, and if you assume that women are less than men, then another hump for men, and there are 25 of each, so there wouldn't be necessarily more men than women, and then it would skew all the way off to the right. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
And this is if there were no gender gap. So this would just be a purely right-skewed distribution, where you have a long tail to the right. Now if you assume that there's some gender gap, then you might have two humps here, which would be a bimodal distribution. So if you assume there's some gender gap, this is part c right here, then maybe there's one hump for women, and if you assume that women are less than men, then another hump for men, and there are 25 of each, so there wouldn't be necessarily more men than women, and then it would skew all the way off to the right. And in fact, I think there would probably be a chance that you have this other notion here, where you have these super CEOs or mega CEOs who make millions, while most CEOs probably just make, I'll put it in quotation marks, a few hundred thousand dollars, while there's a small subset that are way off many standard deviations to the right. So it could even be a trimodal distribution here. So that's choice c. And so far, choice a looks like the best candidate for a pure, or the closest to being a normal distribution. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
So if you assume there's some gender gap, this is part c right here, then maybe there's one hump for women, and if you assume that women are less than men, then another hump for men, and there are 25 of each, so there wouldn't be necessarily more men than women, and then it would skew all the way off to the right. And in fact, I think there would probably be a chance that you have this other notion here, where you have these super CEOs or mega CEOs who make millions, while most CEOs probably just make, I'll put it in quotation marks, a few hundred thousand dollars, while there's a small subset that are way off many standard deviations to the right. So it could even be a trimodal distribution here. So that's choice c. And so far, choice a looks like the best candidate for a pure, or the closest to being a normal distribution. Let's see what choice d is. The dates of 100 pennies taken from a cash drawer in a convenience store. 100 pennies. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
So that's choice c. And so far, choice a looks like the best candidate for a pure, or the closest to being a normal distribution. Let's see what choice d is. The dates of 100 pennies taken from a cash drawer in a convenience store. 100 pennies. So that's actually an interesting experiment. But I would guess, and once again, this is really a question where I get to express my feelings about these things. As long as your answer is reasonable, I would say that it is right. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
100 pennies. So that's actually an interesting experiment. But I would guess, and once again, this is really a question where I get to express my feelings about these things. As long as your answer is reasonable, I would say that it is right. Most pennies are newer pennies, because they go out of commission, they get traded out, they get worn out as they age, they get lost, or they get pressed at the little tourist place into those little souvenir things. I'm not even sure if that's legal, if you can do that to money legally. So my guess is that if you were to plot it, you would have a ton of pennies that are within the last few years. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
As long as your answer is reasonable, I would say that it is right. Most pennies are newer pennies, because they go out of commission, they get traded out, they get worn out as they age, they get lost, or they get pressed at the little tourist place into those little souvenir things. I'm not even sure if that's legal, if you can do that to money legally. So my guess is that if you were to plot it, you would have a ton of pennies that are within the last few years. So the dates of 100 pennies, not their age. So if we're sitting here in 2000, so if this is 2010, I would guess that right now you're not going to find any 2010 pennies, but you're probably going to find a ton of 2009 pennies, and then it probably just goes down from there. And of course, you're not going to find pennies that are older than, say, the United States, or before they even started printing pennies. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
So my guess is that if you were to plot it, you would have a ton of pennies that are within the last few years. So the dates of 100 pennies, not their age. So if we're sitting here in 2000, so if this is 2010, I would guess that right now you're not going to find any 2010 pennies, but you're probably going to find a ton of 2009 pennies, and then it probably just goes down from there. And of course, you're not going to find pennies that are older than, say, the United States, or before they even started printing pennies. So obviously this tail isn't going to go to the left forever, but my guess is you're going to have a left skewed distribution. Where you have the bulk of the distribution on the right, but the tail goes off to the left. That's why it's called a left skewed distribution. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
And of course, you're not going to find pennies that are older than, say, the United States, or before they even started printing pennies. So obviously this tail isn't going to go to the left forever, but my guess is you're going to have a left skewed distribution. Where you have the bulk of the distribution on the right, but the tail goes off to the left. That's why it's called a left skewed distribution. Sometimes this is called a negatively skewed distribution, and similarly this right skewed distribution could also, or this right skewed distribution, sometimes it's called positively skewed. And if you have only one hump, you don't have a multimodal distribution like this, in a left skewed distribution, your mean is going to be to the left of your median. So in this case, maybe your median might be someplace over here. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
That's why it's called a left skewed distribution. Sometimes this is called a negatively skewed distribution, and similarly this right skewed distribution could also, or this right skewed distribution, sometimes it's called positively skewed. And if you have only one hump, you don't have a multimodal distribution like this, in a left skewed distribution, your mean is going to be to the left of your median. So in this case, maybe your median might be someplace over here. But since you have this long tail to the left, your mean might be someplace over here. And likewise, in this distribution, your median, your middle value, might be someplace like this. But because it's right skewed, and for the most part it only has one big hump, this hump won't change things too much because it's small, your mean is going to be to the right of it. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
So in this case, maybe your median might be someplace over here. But since you have this long tail to the left, your mean might be someplace over here. And likewise, in this distribution, your median, your middle value, might be someplace like this. But because it's right skewed, and for the most part it only has one big hump, this hump won't change things too much because it's small, your mean is going to be to the right of it. So that's another reason why it's called a right skewed or positively skewed distribution. So to answer the question, these are my feelings about all of them, but I would say, the other choices explain why you believe they would not follow, well they said, which of the following data sets is most likely to be normally distributed? Well, I would say choice A, but it's really a matter of opinion, at least in this question. | ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3 |
If all four numbers match the four winning numbers, regardless of order, the player wins. What is the probability that the winning numbers are 3, 15, 46, and 49? So the way to think about this problem, they say that we're going to choose four numbers from 60. So one way to think about it is how many different outcomes are there if we choose four numbers out of 60? Now, this is equivalent to saying how many combinations are there if we have 60 items? In this case, we have 60 numbers, and we are going to choose four. And we don't care about the order. | Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So one way to think about it is how many different outcomes are there if we choose four numbers out of 60? Now, this is equivalent to saying how many combinations are there if we have 60 items? In this case, we have 60 numbers, and we are going to choose four. And we don't care about the order. That's why we're dealing with combinations, not permutations. We don't care about the order. How many different groups of four can we pick out of 60? | Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And we don't care about the order. That's why we're dealing with combinations, not permutations. We don't care about the order. How many different groups of four can we pick out of 60? We don't care what order we pick them in. We've seen in previous videos that there is a formula here, but it's important to understand the reasoning behind the formula. I'll write the formula here, but then we'll think about what it's actually saying. | Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3 |
How many different groups of four can we pick out of 60? We don't care what order we pick them in. We've seen in previous videos that there is a formula here, but it's important to understand the reasoning behind the formula. I'll write the formula here, but then we'll think about what it's actually saying. This is 60 factorial over 60 minus 4 factorial divided also by 4 factorial, or in the denominator, multiplied by 4 factorial. This is the formula right here. What this is really saying, this part right here, 60 factorial divided by 60 minus 4 factorial, that's just 60 times 59 times 58 times 57. | Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3 |
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