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If we were to go all the way to free throw number 10, so I'm just skipping a bunch, we're going to get some very, very, very small fraction that have made all 10. It's essentially going to be 75% times 75% times 75%, 10 times. 75% being multiplied repeatedly 10 times. So this is going to be what we're left off with, is going to be 75% times 75%. And let me copy and paste this, just so it doesn't take forever. So copy and then paste it. So times out, but the multiplication signs later. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
So this is going to be what we're left off with, is going to be 75% times 75%. And let me copy and paste this, just so it doesn't take forever. So copy and then paste it. So times out, but the multiplication signs later. So that's 4, that's 6, that's 8, and then that is 10. Right over there, and let me throw the multiplication signs in there. So times, times, times, times. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
So times out, but the multiplication signs later. So that's 4, that's 6, that's 8, and then that is 10. Right over there, and let me throw the multiplication signs in there. So times, times, times, times. So this little fraction that made all 10 of them is going to be equal to this value right over here. 75%. So that's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
So times, times, times, times. So this little fraction that made all 10 of them is going to be equal to this value right over here. 75%. So that's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 75% being repeatedly multiplied 10 times. Now this would obviously take me forever to do it by hand, and even on a calculator if I were to punch all of this in I might make a mistake. But lucky for us, there is a mathematical operator that is essentially repeated multiplication, and that's taking an exponent. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
So that's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 75% being repeatedly multiplied 10 times. Now this would obviously take me forever to do it by hand, and even on a calculator if I were to punch all of this in I might make a mistake. But lucky for us, there is a mathematical operator that is essentially repeated multiplication, and that's taking an exponent. So another way of writing that right over there, we could write that as 75% to the 10th power, repeatedly multiplying 75% 10 times. These are the same expression. And 75%, the word percent, literally means per hundred. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
But lucky for us, there is a mathematical operator that is essentially repeated multiplication, and that's taking an exponent. So another way of writing that right over there, we could write that as 75% to the 10th power, repeatedly multiplying 75% 10 times. These are the same expression. And 75%, the word percent, literally means per hundred. You might recognize the root word cent from things like century, 100 years in a century, 100 cents in a dollar. So this literally means per hundred. So we could write this as 75 over 100 to the 10th power, which is the same thing as 0.75 to the 10th power. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
And 75%, the word percent, literally means per hundred. You might recognize the root word cent from things like century, 100 years in a century, 100 cents in a dollar. So this literally means per hundred. So we could write this as 75 over 100 to the 10th power, which is the same thing as 0.75 to the 10th power. Now let's get our calculator out and see what this evaluates to. So 0.75 to the 10th power gets us to 0.056. And I'll just round to the nearest hundredths. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
So we could write this as 75 over 100 to the 10th power, which is the same thing as 0.75 to the 10th power. Now let's get our calculator out and see what this evaluates to. So 0.75 to the 10th power gets us to 0.056. And I'll just round to the nearest hundredths. So if we round to the nearest hundredths, it gets us to 0.06. So this is roughly equal to, if we round to the nearest hundredths, 0.06, which is equal to roughly, when we round, a 6% probability of making 10 free throws in a row, which even though you have quite a high free throw percentage, this is not that high of a probability. It's a little bit better than a 1 in 20 chance. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
And I'll just round to the nearest hundredths. So if we round to the nearest hundredths, it gets us to 0.06. So this is roughly equal to, if we round to the nearest hundredths, 0.06, which is equal to roughly, when we round, a 6% probability of making 10 free throws in a row, which even though you have quite a high free throw percentage, this is not that high of a probability. It's a little bit better than a 1 in 20 chance. Now what I want to throw out there for everyone else watching this is to think about how we can make a general statement about anybody. If anybody has some free throw percentage and they want to say, what's the probability of making 10 in a row? How can we say that? | Free throwing probability Probability and Statistics Khan Academy.mp3 |
It's a little bit better than a 1 in 20 chance. Now what I want to throw out there for everyone else watching this is to think about how we can make a general statement about anybody. If anybody has some free throw percentage and they want to say, what's the probability of making 10 in a row? How can we say that? Well, I think you saw the pattern right over here. The probability of making, let's call it n, where n is the number of free throws we care about, n free throws in a row for somebody, and we're not just talking about LeBron here, it's going to be their free throw percentage, in this case LeBron's was 75%, to the number of free throws that we want to get in a row, so to the nth power. So for example, you might want to play around with your own free throw percentage. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
How can we say that? Well, I think you saw the pattern right over here. The probability of making, let's call it n, where n is the number of free throws we care about, n free throws in a row for somebody, and we're not just talking about LeBron here, it's going to be their free throw percentage, in this case LeBron's was 75%, to the number of free throws that we want to get in a row, so to the nth power. So for example, you might want to play around with your own free throw percentage. If your free throw percentage, let's say it's 60%, which is the same thing as 0.6. So let's say you have a 60% free throw percentage and you want to see your probability of getting 5 in a row, you would take that to the 5th power. And you'd get what looks like, if you round to the nearest hundredths, it would be about 8%. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
The first graph shows the relationship between test grades and the amount of time the student spent studying. So this is study time on this axis, and this is the test grade on this axis. And the second graph shows the relationship between test grades and shoe size. So shoe size on this axis, and then test grade. Choose the best description of the relationship between the graphs. So first, before even looking at these, let's just look at these. Before looking at the explanations, let's look at the actual graphs. | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
So shoe size on this axis, and then test grade. Choose the best description of the relationship between the graphs. So first, before even looking at these, let's just look at these. Before looking at the explanations, let's look at the actual graphs. So this one on the left right over here, it looks like there is a positive linear relationship right over here. I could almost fit a line that would go just like that. And it makes sense that there would be. | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
Before looking at the explanations, let's look at the actual graphs. So this one on the left right over here, it looks like there is a positive linear relationship right over here. I could almost fit a line that would go just like that. And it makes sense that there would be. That the more time that you spend studying, that the better score that you would get. Now, for a certain amount of time studying, some people might do better than others, but it does seem like there's this relationship. Here, it doesn't seem like there's really much of a relationship. | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
And it makes sense that there would be. That the more time that you spend studying, that the better score that you would get. Now, for a certain amount of time studying, some people might do better than others, but it does seem like there's this relationship. Here, it doesn't seem like there's really much of a relationship. You see the shoe sizes. For a given shoe size, some people do not so well, and some people do very well. For some people, someone with a size 10 and 1 half, it looks like. | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
Here, it doesn't seem like there's really much of a relationship. You see the shoe sizes. For a given shoe size, some people do not so well, and some people do very well. For some people, someone with a size 10 and 1 half, it looks like. Yeah, a size 10 and 1 half, someone, it looks like they flunked the exam. Someone else looks like they got an A minus or a B plus on the exam. And it really would be hard to somehow fit a line here. | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
For some people, someone with a size 10 and 1 half, it looks like. Yeah, a size 10 and 1 half, someone, it looks like they flunked the exam. Someone else looks like they got an A minus or a B plus on the exam. And it really would be hard to somehow fit a line here. No matter how you draw a line, it doesn't seem like it would really fit some type of a, these dots don't seem to form a trend. So let's see which of these choices apply. There's a negative linear relationship between study time and score. | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
And it really would be hard to somehow fit a line here. No matter how you draw a line, it doesn't seem like it would really fit some type of a, these dots don't seem to form a trend. So let's see which of these choices apply. There's a negative linear relationship between study time and score. No, that's not true. It looks like there's a positive linear relationship. The more you study, the better your score would be. | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
There's a negative linear relationship between study time and score. No, that's not true. It looks like there's a positive linear relationship. The more you study, the better your score would be. A negative linear relationship would trend downwards like that. There is a nonlinear relationship between study time and score, and a negative linear relationship between shoe size and score. Well, that doesn't seem right either. | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
The more you study, the better your score would be. A negative linear relationship would trend downwards like that. There is a nonlinear relationship between study time and score, and a negative linear relationship between shoe size and score. Well, that doesn't seem right either. A nonlinear relationship, it would not be easy to fit a line to it. And this one seems like a line would be very reasonable. And there isn't any type of, it doesn't seem like there's any type of relationship between shoe size and score. | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
Well, that doesn't seem right either. A nonlinear relationship, it would not be easy to fit a line to it. And this one seems like a line would be very reasonable. And there isn't any type of, it doesn't seem like there's any type of relationship between shoe size and score. So I wouldn't pick this one either. There is a positive linear relationship between study time and score. That's right. | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
And there isn't any type of, it doesn't seem like there's any type of relationship between shoe size and score. So I wouldn't pick this one either. There is a positive linear relationship between study time and score. That's right. And no relationship between shoe size and score. Well, I'm going to go with that one. Both graphs show positive linear trends of approximately equal strength. | Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3 |
What I want to do in this video is review much of what we've already talked about and then hopefully build some of the intuition on why we divide by n minus 1 if we want to have an unbiased estimate of the population variance when we're calculating the sample variance. So let's think about a population. So let's say this is the population right over here and it is of size capital N. And we also have a sample of that population. So a sample of that population and at its size we have lowercase n data points. So let's think about all of the parameters and statistics that we know about so far. So the first is the idea of the mean. So if we're trying to calculate the mean for the population, is that going to be a parameter or a statistic? | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
So a sample of that population and at its size we have lowercase n data points. So let's think about all of the parameters and statistics that we know about so far. So the first is the idea of the mean. So if we're trying to calculate the mean for the population, is that going to be a parameter or a statistic? Well, when we're trying to calculate it on the population, we are calculating a parameter. So let me write this down. So this is going to be, so for the population, we are calculating a parameter. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
So if we're trying to calculate the mean for the population, is that going to be a parameter or a statistic? Well, when we're trying to calculate it on the population, we are calculating a parameter. So let me write this down. So this is going to be, so for the population, we are calculating a parameter. And when we attempt to calculate something for a sample, we would call that a statistic. So how do we think about the mean for a population? Well, first of all, we denote it with the Greek letter mu. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
So this is going to be, so for the population, we are calculating a parameter. And when we attempt to calculate something for a sample, we would call that a statistic. So how do we think about the mean for a population? Well, first of all, we denote it with the Greek letter mu. And we essentially take every data point in our population. So we take the sum of every data point. So we start at the first data point and we go all the way to the capital Nth data point. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
Well, first of all, we denote it with the Greek letter mu. And we essentially take every data point in our population. So we take the sum of every data point. So we start at the first data point and we go all the way to the capital Nth data point. So every data point we add up. So this is the ith data point. So X sub 1 plus X sub 2 all the way to X sub capital N. And then we divide by the total number of data points we have. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
So we start at the first data point and we go all the way to the capital Nth data point. So every data point we add up. So this is the ith data point. So X sub 1 plus X sub 2 all the way to X sub capital N. And then we divide by the total number of data points we have. Well, how do we calculate the sample mean? Well, the sample mean, we do a very similar thing with the sample. And we denote it with an X with a bar over it. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
So X sub 1 plus X sub 2 all the way to X sub capital N. And then we divide by the total number of data points we have. Well, how do we calculate the sample mean? Well, the sample mean, we do a very similar thing with the sample. And we denote it with an X with a bar over it. And that's going to be taking every data point in the sample, so going up to lowercase n, adding them up. So these are the sum of all the data points in our sample. And then dividing by the number of data points that we actually had. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
And we denote it with an X with a bar over it. And that's going to be taking every data point in the sample, so going up to lowercase n, adding them up. So these are the sum of all the data points in our sample. And then dividing by the number of data points that we actually had. Now, the other thing that we're trying to calculate for the population, which was a parameter, and then we'll also try to calculate it for the sample and estimate it for the population, was the variance, which was a measure of how dispersed or how much the data points vary from the mean. So let's write variance right over here. And how do we denote and calculate variance for a population? | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
And then dividing by the number of data points that we actually had. Now, the other thing that we're trying to calculate for the population, which was a parameter, and then we'll also try to calculate it for the sample and estimate it for the population, was the variance, which was a measure of how dispersed or how much the data points vary from the mean. So let's write variance right over here. And how do we denote and calculate variance for a population? Well, for a population, we'd say that the variance, we use the Greek letter sigma squared, is equal to, and you could view it as the mean of the squared distances from the population mean. But what we do is we take, for each data point, so I equal 1 all the way to N, we take that data point, subtract from it the population mean. So if you want to calculate this, you'd want to figure this out. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
And how do we denote and calculate variance for a population? Well, for a population, we'd say that the variance, we use the Greek letter sigma squared, is equal to, and you could view it as the mean of the squared distances from the population mean. But what we do is we take, for each data point, so I equal 1 all the way to N, we take that data point, subtract from it the population mean. So if you want to calculate this, you'd want to figure this out. Or that's one way to do it. We'll see there's other ways to do it, where you can kind of calculate them at the same time. But you would, the easiest or the most intuitive, calculate this first. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
So if you want to calculate this, you'd want to figure this out. Or that's one way to do it. We'll see there's other ways to do it, where you can kind of calculate them at the same time. But you would, the easiest or the most intuitive, calculate this first. And for each of the data points, take the data point and subtract it from that. Subtract the mean from that. Square it. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
But you would, the easiest or the most intuitive, calculate this first. And for each of the data points, take the data point and subtract it from that. Subtract the mean from that. Square it. And then divide by the total number of data points you have. Now we get to the interesting part, sample variance. There are several ways where, when people talk about sample variance, there are several, I guess, tools in their toolkits, or there are several ways to calculate it. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
Square it. And then divide by the total number of data points you have. Now we get to the interesting part, sample variance. There are several ways where, when people talk about sample variance, there are several, I guess, tools in their toolkits, or there are several ways to calculate it. One way is the biased sample variance, the non-unbiased estimator of the population variance. And that's denoted, usually denoted, by S with a subscript N. And what is the biased estimator? How would we calculate it? | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
There are several ways where, when people talk about sample variance, there are several, I guess, tools in their toolkits, or there are several ways to calculate it. One way is the biased sample variance, the non-unbiased estimator of the population variance. And that's denoted, usually denoted, by S with a subscript N. And what is the biased estimator? How would we calculate it? Well, we would calculate it very similar to how we calculated the variance right over here, but we would do it for our sample, not our population. So for every data point in our sample, so we have N of them, we take that data point and from it we subtract our sample mean, we subtract our sample mean, square it, and then divide by the number of data points that we have. But we already talked about in the last video, how would we find what is our best unbiased estimate of the population variance? | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
How would we calculate it? Well, we would calculate it very similar to how we calculated the variance right over here, but we would do it for our sample, not our population. So for every data point in our sample, so we have N of them, we take that data point and from it we subtract our sample mean, we subtract our sample mean, square it, and then divide by the number of data points that we have. But we already talked about in the last video, how would we find what is our best unbiased estimate of the population variance? This is usually what we're trying to get at. We're trying to find an unbiased estimate of the population variance. Well, in the last video we talked about that if we want to have an unbiased estimate, and here in this video I want to give you a sense of the intuition why, we would take the sum, so we're going to go through every data point in our sample, we're going to take that data point, subtract from it the sample mean, square that, but instead of dividing by N, we divide by N minus 1. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
But we already talked about in the last video, how would we find what is our best unbiased estimate of the population variance? This is usually what we're trying to get at. We're trying to find an unbiased estimate of the population variance. Well, in the last video we talked about that if we want to have an unbiased estimate, and here in this video I want to give you a sense of the intuition why, we would take the sum, so we're going to go through every data point in our sample, we're going to take that data point, subtract from it the sample mean, square that, but instead of dividing by N, we divide by N minus 1. We're dividing by a smaller number. And when you divide by a smaller number, you're going to get a larger value. So this is going to be larger, this is going to be larger, this is going to be smaller. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
Well, in the last video we talked about that if we want to have an unbiased estimate, and here in this video I want to give you a sense of the intuition why, we would take the sum, so we're going to go through every data point in our sample, we're going to take that data point, subtract from it the sample mean, square that, but instead of dividing by N, we divide by N minus 1. We're dividing by a smaller number. And when you divide by a smaller number, you're going to get a larger value. So this is going to be larger, this is going to be larger, this is going to be smaller. And this one we refer to the unbiased estimate. And this one we refer to the biased estimate. If people just write this, they're talking about the sample variance, it's a good idea to clarify which one they're talking about, but if you had to guess and people give you no further information, they're probably talking about the unbiased estimate of the variance. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
So this is going to be larger, this is going to be larger, this is going to be smaller. And this one we refer to the unbiased estimate. And this one we refer to the biased estimate. If people just write this, they're talking about the sample variance, it's a good idea to clarify which one they're talking about, but if you had to guess and people give you no further information, they're probably talking about the unbiased estimate of the variance. So you'd probably divide by N minus 1. But let's think about why this estimate would be biased, and why we might want to have an estimate like this that is larger, and then maybe in the future we could have a computer program or something that really makes us feel better that dividing by N minus 1 gives us a better estimate of the true population variance. So let's imagine all of the data in a population, and I'm just going to plot them on a number line. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
If people just write this, they're talking about the sample variance, it's a good idea to clarify which one they're talking about, but if you had to guess and people give you no further information, they're probably talking about the unbiased estimate of the variance. So you'd probably divide by N minus 1. But let's think about why this estimate would be biased, and why we might want to have an estimate like this that is larger, and then maybe in the future we could have a computer program or something that really makes us feel better that dividing by N minus 1 gives us a better estimate of the true population variance. So let's imagine all of the data in a population, and I'm just going to plot them on a number line. All the data. So this is my number line, this is my number line, and let me plot all of the data points in my population. So this is some data, this is some data, here's some data, and here is some data here, and I can just do as many points as I want. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
So let's imagine all of the data in a population, and I'm just going to plot them on a number line. All the data. So this is my number line, this is my number line, and let me plot all of the data points in my population. So this is some data, this is some data, here's some data, and here is some data here, and I can just do as many points as I want. So these are just points on the number line. Now let's say I take a sample of this. So this is my entire population. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
So this is some data, this is some data, here's some data, and here is some data here, and I can just do as many points as I want. So these are just points on the number line. Now let's say I take a sample of this. So this is my entire population. So let's see how many I have. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. So in this case, what would be my big N? | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
So this is my entire population. So let's see how many I have. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. So in this case, what would be my big N? My big N would be 14. Now let's say I take a sample, a lowercase N of, let's say my sample size is 3. I could take, well, before I even think about that, let's think about roughly where the mean of this population would sit. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
So in this case, what would be my big N? My big N would be 14. Now let's say I take a sample, a lowercase N of, let's say my sample size is 3. I could take, well, before I even think about that, let's think about roughly where the mean of this population would sit. So the way I drew it, and I'm not going to calculate it exactly, it looks like the mean might sit someplace roughly right over here. So the mean, the true population mean, the parameter is going to sit right over here. Now let's think about what happens when we sample. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
I could take, well, before I even think about that, let's think about roughly where the mean of this population would sit. So the way I drew it, and I'm not going to calculate it exactly, it looks like the mean might sit someplace roughly right over here. So the mean, the true population mean, the parameter is going to sit right over here. Now let's think about what happens when we sample. And I'm going to do just a very small sample size just to give us the intuition, but this is true of any sample size. So let's say we have sample size of 3. So there is some possibility when we take our sample size of 3 that we happen to sample it in a way that our sample mean is pretty close to our population mean. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
Now let's think about what happens when we sample. And I'm going to do just a very small sample size just to give us the intuition, but this is true of any sample size. So let's say we have sample size of 3. So there is some possibility when we take our sample size of 3 that we happen to sample it in a way that our sample mean is pretty close to our population mean. So for example, if we sample to that point, that point, and that point, I could imagine our sample mean might actually sit pretty close to our population mean. But there's a distinct possibility that maybe when I take a sample, I sample that, that, and that. And the key idea here is when you take a sample, your sample mean is always going to sit within your sample. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
So there is some possibility when we take our sample size of 3 that we happen to sample it in a way that our sample mean is pretty close to our population mean. So for example, if we sample to that point, that point, and that point, I could imagine our sample mean might actually sit pretty close to our population mean. But there's a distinct possibility that maybe when I take a sample, I sample that, that, and that. And the key idea here is when you take a sample, your sample mean is always going to sit within your sample. And so there is a possibility that when you take your sample, your mean could even be outside of the sample. And so in this situation, and this is just to give you an intuition, so here your sample mean is going to be sitting someplace in there. And so if you were to just calculate the distance from each of these points to the sample mean, so this distance, that distance, and you square it, and you were to divide by the number of data points you have, this is going to be a much lower estimate than the true variance from the actual population mean, where these things are much, much, much further. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
And the key idea here is when you take a sample, your sample mean is always going to sit within your sample. And so there is a possibility that when you take your sample, your mean could even be outside of the sample. And so in this situation, and this is just to give you an intuition, so here your sample mean is going to be sitting someplace in there. And so if you were to just calculate the distance from each of these points to the sample mean, so this distance, that distance, and you square it, and you were to divide by the number of data points you have, this is going to be a much lower estimate than the true variance from the actual population mean, where these things are much, much, much further. Now you're always not going to have the true population mean outside of your sample, but it's possible that you do. So in general, when you just take your points, find the square to distance to your sample mean, which is always going to sit inside of your data, even though the true population mean could be outside of it, or it could be at one end of your data, however you might want to think about it, you are likely to be underestimating the true population variance. So this right over here is an underestimate. | Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3 |
Find the probability of getting exactly two heads when flipping three coins. So let's think about the sample space here. Let's think about all of the possible outcomes. So I could get all heads. So on flip one I get a head, flip two I get a head, flip three I get a head. I could get two heads and then a tail. I could get heads, tail, heads. | Example All the ways you can flip a coin Probability and Statistics Khan Academy.mp3 |
So I could get all heads. So on flip one I get a head, flip two I get a head, flip three I get a head. I could get two heads and then a tail. I could get heads, tail, heads. Or I could get heads, tails, tails. I could get tails, heads, heads. I could get tail, heads, tails. | Example All the ways you can flip a coin Probability and Statistics Khan Academy.mp3 |
I could get heads, tail, heads. Or I could get heads, tails, tails. I could get tails, heads, heads. I could get tail, heads, tails. I could get tails, tails, heads. Or I could get tails, tails, and tails. These are all the different ways that I could flip three coins. | Example All the ways you can flip a coin Probability and Statistics Khan Academy.mp3 |
I could get tail, heads, tails. I could get tails, tails, heads. Or I could get tails, tails, and tails. These are all the different ways that I could flip three coins. And you could maybe say that this is the first flip, the second flip, and the third flip. Now, so this right over here is the sample space. There's eight possible outcomes. | Example All the ways you can flip a coin Probability and Statistics Khan Academy.mp3 |
These are all the different ways that I could flip three coins. And you could maybe say that this is the first flip, the second flip, and the third flip. Now, so this right over here is the sample space. There's eight possible outcomes. Let me write this. The probability of exactly two heads. I'll say h is there for short. | Example All the ways you can flip a coin Probability and Statistics Khan Academy.mp3 |
There's eight possible outcomes. Let me write this. The probability of exactly two heads. I'll say h is there for short. The probability of exactly two heads. Well, what is the size of our sample space? I have eight possible outcomes. | Example All the ways you can flip a coin Probability and Statistics Khan Academy.mp3 |
I'll say h is there for short. The probability of exactly two heads. Well, what is the size of our sample space? I have eight possible outcomes. So eight, this is possible outcomes, or the size of our sample space, possible outcomes. And how many of those possible outcomes are associated with this event? You could call this a compound event, because there's more than one outcome that's associated with this. | Example All the ways you can flip a coin Probability and Statistics Khan Academy.mp3 |
I have eight possible outcomes. So eight, this is possible outcomes, or the size of our sample space, possible outcomes. And how many of those possible outcomes are associated with this event? You could call this a compound event, because there's more than one outcome that's associated with this. Well, so let's think about exactly two heads. This is three heads, so it's not exactly two heads. This is exactly two heads right over here. | Example All the ways you can flip a coin Probability and Statistics Khan Academy.mp3 |
You could call this a compound event, because there's more than one outcome that's associated with this. Well, so let's think about exactly two heads. This is three heads, so it's not exactly two heads. This is exactly two heads right over here. This is exactly two heads right over here. There's only one head. This is exactly two heads. | Example All the ways you can flip a coin Probability and Statistics Khan Academy.mp3 |
This is exactly two heads right over here. This is exactly two heads right over here. There's only one head. This is exactly two heads. This is only one head, only one head, no heads. So you have one, two, three of the possible outcomes are associated with this event. So you have three possible outcomes. | Example All the ways you can flip a coin Probability and Statistics Khan Academy.mp3 |
This is exactly two heads. This is only one head, only one head, no heads. So you have one, two, three of the possible outcomes are associated with this event. So you have three possible outcomes. Three outcomes associated with the event. Three outcomes satisfy this event, or are associated with this event. So the probability of getting exactly two heads when flipping three coins is three outcomes satisfying this event over eight possible outcomes. | Example All the ways you can flip a coin Probability and Statistics Khan Academy.mp3 |
And our alternative hypothesis is that the drug just has an effect. We didn't say whether the drug would lower the response time or raise the response time. We just said the drug has an effect that it will not, the mean when you have the drug will not be the same thing as the population mean. And then the null hypothesis is no, your mean with the drug is going to be the same thing as the population mean. It has no effect. In this situation, where we're really just testing to see if it had an effect, whether an extreme positive effect or an extreme negative effect would have both been considered an effect, we did something called a two-tailed test. This is called a two-tailed test. | One-tailed and two-tailed tests Inferential statistics Probability and Statistics Khan Academy.mp3 |
And then the null hypothesis is no, your mean with the drug is going to be the same thing as the population mean. It has no effect. In this situation, where we're really just testing to see if it had an effect, whether an extreme positive effect or an extreme negative effect would have both been considered an effect, we did something called a two-tailed test. This is called a two-tailed test. Because frankly, a super high response time, if you had a response time that was more than 3 standard deviations, that would have also made us likely to reject the null hypothesis. So we were dealing with kind of both tails. You could have done a similar type of hypothesis test with the same experiment, where you only have a one-tailed test. | One-tailed and two-tailed tests Inferential statistics Probability and Statistics Khan Academy.mp3 |
This is called a two-tailed test. Because frankly, a super high response time, if you had a response time that was more than 3 standard deviations, that would have also made us likely to reject the null hypothesis. So we were dealing with kind of both tails. You could have done a similar type of hypothesis test with the same experiment, where you only have a one-tailed test. And the way we could have done that is we still could have had the null hypothesis, we still could have had the null hypothesis be that the drug has no effect. Drug has no effect. Or that the mean with the drug is still going to be 1.2 seconds, our mean response time. | One-tailed and two-tailed tests Inferential statistics Probability and Statistics Khan Academy.mp3 |
You could have done a similar type of hypothesis test with the same experiment, where you only have a one-tailed test. And the way we could have done that is we still could have had the null hypothesis, we still could have had the null hypothesis be that the drug has no effect. Drug has no effect. Or that the mean with the drug is still going to be 1.2 seconds, our mean response time. Now if we wanted to do a one-tailed test, and for some reason we already had maybe a view that this drug would lower response times, then our alternative hypothesis, and just so you get familiar with different types of notation, some books or teachers will write the alternative hypothesis as H1, sometimes they write it as H alternative, either one is fine. If you want to do a one-tailed test, you could say that the drug lowers response time. Or that the mean with the drug is less than 1.2 seconds. | One-tailed and two-tailed tests Inferential statistics Probability and Statistics Khan Academy.mp3 |
Or that the mean with the drug is still going to be 1.2 seconds, our mean response time. Now if we wanted to do a one-tailed test, and for some reason we already had maybe a view that this drug would lower response times, then our alternative hypothesis, and just so you get familiar with different types of notation, some books or teachers will write the alternative hypothesis as H1, sometimes they write it as H alternative, either one is fine. If you want to do a one-tailed test, you could say that the drug lowers response time. Or that the mean with the drug is less than 1.2 seconds. Now if you do a one-tailed test like this, what we're thinking about is, what we want to look at is, we have our sampling distribution. Actually I can just use the drawing that I had up here. You had your sampling distribution of the sample mean. | One-tailed and two-tailed tests Inferential statistics Probability and Statistics Khan Academy.mp3 |
Or that the mean with the drug is less than 1.2 seconds. Now if you do a one-tailed test like this, what we're thinking about is, what we want to look at is, we have our sampling distribution. Actually I can just use the drawing that I had up here. You had your sampling distribution of the sample mean. We know what the mean of that was, it's 1.2 seconds, same as the population mean. We were able to estimate its standard deviation using our sample standard deviation, and that was reasonable because it has a sample size of greater than 30, so we can still deal with a normal distribution for the sampling distribution. And using that, we saw that the result, the sample mean that we got, the 1.05 seconds, is three standard deviations below the mean. | One-tailed and two-tailed tests Inferential statistics Probability and Statistics Khan Academy.mp3 |
You had your sampling distribution of the sample mean. We know what the mean of that was, it's 1.2 seconds, same as the population mean. We were able to estimate its standard deviation using our sample standard deviation, and that was reasonable because it has a sample size of greater than 30, so we can still deal with a normal distribution for the sampling distribution. And using that, we saw that the result, the sample mean that we got, the 1.05 seconds, is three standard deviations below the mean. So if we look at it, let me just redraw it with our new hypothesis test. So this is the sampling distribution. It has a mean right over here at 1.2 seconds. | One-tailed and two-tailed tests Inferential statistics Probability and Statistics Khan Academy.mp3 |
And using that, we saw that the result, the sample mean that we got, the 1.05 seconds, is three standard deviations below the mean. So if we look at it, let me just redraw it with our new hypothesis test. So this is the sampling distribution. It has a mean right over here at 1.2 seconds. And the result we got was three standard deviations below the mean. One, two, three standard deviations below the mean. That was what our 1.05 seconds were. | One-tailed and two-tailed tests Inferential statistics Probability and Statistics Khan Academy.mp3 |
It has a mean right over here at 1.2 seconds. And the result we got was three standard deviations below the mean. One, two, three standard deviations below the mean. That was what our 1.05 seconds were. So when you set it up like this, where you're not just saying that the drug has an effect, in that case, and that was the last video, you'd look at both tails. But here we're saying we only care, does the drug lower our response time? And just like we did before, you say, okay, let's say the drug doesn't lower our response time. | One-tailed and two-tailed tests Inferential statistics Probability and Statistics Khan Academy.mp3 |
That was what our 1.05 seconds were. So when you set it up like this, where you're not just saying that the drug has an effect, in that case, and that was the last video, you'd look at both tails. But here we're saying we only care, does the drug lower our response time? And just like we did before, you say, okay, let's say the drug doesn't lower our response time. If the drug doesn't lower our response time, what was the probability, or what is the probability of getting a lowering this extreme or more extreme? So here it will only be one of the tails that we consider when we set our alternative hypothesis like that, that we think it lowers. So if our null hypothesis is true, the probability of getting a result more extreme than 1.05 seconds, now we are only considering this tail right over here. | One-tailed and two-tailed tests Inferential statistics Probability and Statistics Khan Academy.mp3 |
And just like we did before, you say, okay, let's say the drug doesn't lower our response time. If the drug doesn't lower our response time, what was the probability, or what is the probability of getting a lowering this extreme or more extreme? So here it will only be one of the tails that we consider when we set our alternative hypothesis like that, that we think it lowers. So if our null hypothesis is true, the probability of getting a result more extreme than 1.05 seconds, now we are only considering this tail right over here. Let me put it this way. More extreme than 1.05 seconds, or let me say lower, because in the last video we cared about more extreme, because even a really high result would have said, okay, the mean is definitely not 1.2 seconds. But in this case we care about means that are lower. | One-tailed and two-tailed tests Inferential statistics Probability and Statistics Khan Academy.mp3 |
So if our null hypothesis is true, the probability of getting a result more extreme than 1.05 seconds, now we are only considering this tail right over here. Let me put it this way. More extreme than 1.05 seconds, or let me say lower, because in the last video we cared about more extreme, because even a really high result would have said, okay, the mean is definitely not 1.2 seconds. But in this case we care about means that are lower. So now we care about the probability of a result lower than 1.05 seconds. That's the same thing as getting a sample from the sampling distribution that's more than three standard deviations below the mean. And in this case we are only going to consider the area in this one tail. | One-tailed and two-tailed tests Inferential statistics Probability and Statistics Khan Academy.mp3 |
But in this case we care about means that are lower. So now we care about the probability of a result lower than 1.05 seconds. That's the same thing as getting a sample from the sampling distribution that's more than three standard deviations below the mean. And in this case we are only going to consider the area in this one tail. So this right here would be a one-tail test, where we only care about one direction below the mean. And if you look at the one-tail test, this area over here, we saw last time that both of these areas combined are 0.3%. But if you're only considering one of these areas, if you're only considering this one over here, it's going to be half of that, because the normal distribution is symmetric. | One-tailed and two-tailed tests Inferential statistics Probability and Statistics Khan Academy.mp3 |
And in this case we are only going to consider the area in this one tail. So this right here would be a one-tail test, where we only care about one direction below the mean. And if you look at the one-tail test, this area over here, we saw last time that both of these areas combined are 0.3%. But if you're only considering one of these areas, if you're only considering this one over here, it's going to be half of that, because the normal distribution is symmetric. So it's going to be 0.13%. So this one right here is going to be 0.15%. Or if you were to express it as a decimal, this is going to be 0.0015. | One-tailed and two-tailed tests Inferential statistics Probability and Statistics Khan Academy.mp3 |
But if you're only considering one of these areas, if you're only considering this one over here, it's going to be half of that, because the normal distribution is symmetric. So it's going to be 0.13%. So this one right here is going to be 0.15%. Or if you were to express it as a decimal, this is going to be 0.0015. So once again, if you set up your hypotheses like this, you would have said, if your null hypothesis is correct, there would have only been a 0.15% chance of getting a result lower than the result we got. So that would be very unlikely, so we will reject the null hypothesis and go with the alternative. And in this situation, your p-value is going to be the 0.0015. | One-tailed and two-tailed tests Inferential statistics Probability and Statistics Khan Academy.mp3 |
So there's going to be two bins of balls. So you're gonna have two bins of balls. One of them's gonna have 56 balls in it. So 56 in one bin. And then another bin is going to have 46 balls in it. So there are 46 balls in this bin right over here. And so what they're gonna do is they're gonna pick five balls from this bin right over here. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So 56 in one bin. And then another bin is going to have 46 balls in it. So there are 46 balls in this bin right over here. And so what they're gonna do is they're gonna pick five balls from this bin right over here. And you have to get the exact numbers of those five balls. It can be in any order. So let me just draw them. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And so what they're gonna do is they're gonna pick five balls from this bin right over here. And you have to get the exact numbers of those five balls. It can be in any order. So let me just draw them. So it's one ball. I'll shade it so it looks like a ball. Two balls. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So let me just draw them. So it's one ball. I'll shade it so it looks like a ball. Two balls. Three balls. Four balls. And five balls that they're going to pick. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Two balls. Three balls. Four balls. And five balls that they're going to pick. And you just have to get the numbers in any order. So this is from a bin of 56. From a bin of 56. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And five balls that they're going to pick. And you just have to get the numbers in any order. So this is from a bin of 56. From a bin of 56. And then you have to get the mega ball right. And then they're gonna just pick one ball from there, which they call the mega ball. They're gonna pick one ball from there. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
From a bin of 56. And then you have to get the mega ball right. And then they're gonna just pick one ball from there, which they call the mega ball. They're gonna pick one ball from there. And obviously this is just going to be picked, this is gonna be one of 46. So from a bin of 46. And so to figure out the probability of winning, it's essentially going to be one of all of the possible, all of the possibilities of numbers that you might be able to pick. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
They're gonna pick one ball from there. And obviously this is just going to be picked, this is gonna be one of 46. So from a bin of 46. And so to figure out the probability of winning, it's essentially going to be one of all of the possible, all of the possibilities of numbers that you might be able to pick. So essentially all of the combinations of the white balls times the 46 possibilities that you might get for the mega ball. So to think about the combinations for the white balls, there's a couple of ways you could do it. If you are used to thinking in combinatorics terms, it would essentially saying, well, out of a set of 56 things, I am going to choose five of them. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And so to figure out the probability of winning, it's essentially going to be one of all of the possible, all of the possibilities of numbers that you might be able to pick. So essentially all of the combinations of the white balls times the 46 possibilities that you might get for the mega ball. So to think about the combinations for the white balls, there's a couple of ways you could do it. If you are used to thinking in combinatorics terms, it would essentially saying, well, out of a set of 56 things, I am going to choose five of them. So this is literally, you could view this as 56 choose five. Or if you wanna think of it in more conceptual terms, the first ball I pick, there's 56 possibilities. Since we're not replacing the ball, the next ball I pick, there's going to be 55 possibilities. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
If you are used to thinking in combinatorics terms, it would essentially saying, well, out of a set of 56 things, I am going to choose five of them. So this is literally, you could view this as 56 choose five. Or if you wanna think of it in more conceptual terms, the first ball I pick, there's 56 possibilities. Since we're not replacing the ball, the next ball I pick, there's going to be 55 possibilities. The ball after that, there's going to be 54 possibilities. Ball after that, there's going to be 53 possibilities. And then the ball after that, there's going to be 52 possibilities. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Since we're not replacing the ball, the next ball I pick, there's going to be 55 possibilities. The ball after that, there's going to be 54 possibilities. Ball after that, there's going to be 53 possibilities. And then the ball after that, there's going to be 52 possibilities. 52, because I've already picked four balls out of that. Now this number right over here, when you multiply it out, this is the number of permutations if I cared about order. So if I got that exact combination. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And then the ball after that, there's going to be 52 possibilities. 52, because I've already picked four balls out of that. Now this number right over here, when you multiply it out, this is the number of permutations if I cared about order. So if I got that exact combination. But to win this, you don't have to write them down in the same order. You just have to get those numbers in any order. And so what you wanna do is you wanna divide this by the number of ways that five things can actually be ordered. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So if I got that exact combination. But to win this, you don't have to write them down in the same order. You just have to get those numbers in any order. And so what you wanna do is you wanna divide this by the number of ways that five things can actually be ordered. So what you wanna do is divide this by the way that five things can be ordered. And if you're ordering five things, the first of the five things can take five different positions. Then the next one will have four positions left. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And so what you wanna do is you wanna divide this by the number of ways that five things can actually be ordered. So what you wanna do is divide this by the way that five things can be ordered. And if you're ordering five things, the first of the five things can take five different positions. Then the next one will have four positions left. Then the one after that will have three positions left. One after that will have two positions. And then the fifth one will be completely determined because you've already placed the other four. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Then the next one will have four positions left. Then the one after that will have three positions left. One after that will have two positions. And then the fifth one will be completely determined because you've already placed the other four. So it's going to have only one position. So when we calculate this part right over here, this will tell us all of the combinations of just the white balls. And so let's calculate that. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And then the fifth one will be completely determined because you've already placed the other four. So it's going to have only one position. So when we calculate this part right over here, this will tell us all of the combinations of just the white balls. And so let's calculate that. So just the white balls, we have 55, sorry, 56 times 55 times 54 times 53 times 52. And we're gonna divide that by five times four times three times two. We don't have to multiply by one, but I'll just do that just to show what we're doing. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And so let's calculate that. So just the white balls, we have 55, sorry, 56 times 55 times 54 times 53 times 52. And we're gonna divide that by five times four times three times two. We don't have to multiply by one, but I'll just do that just to show what we're doing. And then that gives us about 3.8 million. So let me actually take that, let me put that off screen. So let me write that number down. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
We don't have to multiply by one, but I'll just do that just to show what we're doing. And then that gives us about 3.8 million. So let me actually take that, let me put that off screen. So let me write that number down. So this comes out to 3,819,816. So that's the number of possibilities here. So just your odds of picking just the white balls right are going to be one out of this, assuming you only have one entry. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So let me write that number down. So this comes out to 3,819,816. So that's the number of possibilities here. So just your odds of picking just the white balls right are going to be one out of this, assuming you only have one entry. And then there's 46 possibilities for the orange ball. So you're gonna multiply that times 46. And so that's going to get you, so when you multiply it times 46, bring the calculator back. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So just your odds of picking just the white balls right are going to be one out of this, assuming you only have one entry. And then there's 46 possibilities for the orange ball. So you're gonna multiply that times 46. And so that's going to get you, so when you multiply it times 46, bring the calculator back. So we're gonna multiply our previous answer times 46. And it just means my previous answer times 46. I get a little under 176 million. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And so that's going to get you, so when you multiply it times 46, bring the calculator back. So we're gonna multiply our previous answer times 46. And it just means my previous answer times 46. I get a little under 176 million. A little under 176 million. So that is, let me write that number down. So that gives us 175,711,536. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
I get a little under 176 million. A little under 176 million. So that is, let me write that number down. So that gives us 175,711,536. So your odds of winning it with one entry, because this is the number of possibilities and you are essentially for a dollar getting one of those possibilities, your odds of winning is going to be one over this. And to put this in a little bit of context, I looked it up on the internet, what your odds are of actually getting struck by lightning in your lifetime. And so your odds of getting struck by lightning in your lifetime is roughly one in 10,000. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So that gives us 175,711,536. So your odds of winning it with one entry, because this is the number of possibilities and you are essentially for a dollar getting one of those possibilities, your odds of winning is going to be one over this. And to put this in a little bit of context, I looked it up on the internet, what your odds are of actually getting struck by lightning in your lifetime. And so your odds of getting struck by lightning in your lifetime is roughly one in 10,000. One in 10,000 chance of getting struck by lightning in your lifetime. And we can roughly say your odds of getting struck by lightning twice in your lifetime, or another way of saying it is the odds of you and your best friend both independently being struck by lightning when you're not around each other is going to be one in 10,000 times one in 10,000. And so that will get you one in, and we're going to have now eight zeros. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And so your odds of getting struck by lightning in your lifetime is roughly one in 10,000. One in 10,000 chance of getting struck by lightning in your lifetime. And we can roughly say your odds of getting struck by lightning twice in your lifetime, or another way of saying it is the odds of you and your best friend both independently being struck by lightning when you're not around each other is going to be one in 10,000 times one in 10,000. And so that will get you one in, and we're going to have now eight zeros. One, two, three, four, five, six, seven, eight. So that gives you one in 100 million. So you're actually twice almost, this is very rough, you're roughly twice as likely to get struck by lightning twice in your life than to win the mega jackpot. | Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Well, this looks pretty much like a binomial random variable. In fact, I'm pretty confident it is a binomial random variable, and we could just go down the checklist. The outcome of each trial can be a success or failure. 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. | Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3 |
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