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Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3
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Let's say we're trying to understand the relationship between people's height and their weight. So what we do is we go to 10 different people and we measure each of their heights and each of their weights. And so on this scatter plot here, each dot represents a person. So for example, this dot over here represents a person whose height was 60 inches or five feet tall. So that's the point 60 comma, and whose weight, which we have on the y-axis, was 125 pounds. And so when you look at this scatter plot, your eyes naturally see some type of a trend. It seems like, generally speaking, as height increases, weight increases as well.
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Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3
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So for example, this dot over here represents a person whose height was 60 inches or five feet tall. So that's the point 60 comma, and whose weight, which we have on the y-axis, was 125 pounds. And so when you look at this scatter plot, your eyes naturally see some type of a trend. It seems like, generally speaking, as height increases, weight increases as well. But I said generally speaking. You definitely have circumstances where there are taller people who might weigh less. But an interesting question is, can we try to fit a line to this data?
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Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3
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It seems like, generally speaking, as height increases, weight increases as well. But I said generally speaking. You definitely have circumstances where there are taller people who might weigh less. But an interesting question is, can we try to fit a line to this data? And this idea of trying to fit a line as closely as possible to as many of the points as possible is known as linear regression. Now, the most common technique is to try to fit a line that minimizes the squared distance to each of those points. And we're gonna talk more about that in future videos.
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Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3
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But an interesting question is, can we try to fit a line to this data? And this idea of trying to fit a line as closely as possible to as many of the points as possible is known as linear regression. Now, the most common technique is to try to fit a line that minimizes the squared distance to each of those points. And we're gonna talk more about that in future videos. But for now, we wanna get an intuitive feel for that. So if you were to just eyeball it and look at a line like that, you wouldn't think that it would be a particularly good fit. It looks like most of the data sits above the line.
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Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3
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And we're gonna talk more about that in future videos. But for now, we wanna get an intuitive feel for that. So if you were to just eyeball it and look at a line like that, you wouldn't think that it would be a particularly good fit. It looks like most of the data sits above the line. Similarly, something like this also doesn't look that great. Here, most of our data points are sitting below the line. But something like this actually looks very good.
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Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3
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It looks like most of the data sits above the line. Similarly, something like this also doesn't look that great. Here, most of our data points are sitting below the line. But something like this actually looks very good. It looks like it's getting as close as possible to as many of the points as possible. It seems like it's describing this general trend. And so this is the actual regression line.
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Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3
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But something like this actually looks very good. It looks like it's getting as close as possible to as many of the points as possible. It seems like it's describing this general trend. And so this is the actual regression line. And the equation here, we would write as, we would write y with a little hat over it. And that means that we are trying to estimate a y for a given x. It's not always going to be the actual y for a given x because as we see, sometimes the points aren't sitting on the line.
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Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3
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And so this is the actual regression line. And the equation here, we would write as, we would write y with a little hat over it. And that means that we are trying to estimate a y for a given x. It's not always going to be the actual y for a given x because as we see, sometimes the points aren't sitting on the line. But we say y hat is equal to, and our y-intercept for this particular regression line, it is negative 140 plus the slope, 14 over three, times x. Now as we can see, for most of these points, given the x value of those points, the estimate that our regression line gives is different than the actual value. And that difference between the actual and the estimate from the regression line is known as the residual.
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Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3
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It's not always going to be the actual y for a given x because as we see, sometimes the points aren't sitting on the line. But we say y hat is equal to, and our y-intercept for this particular regression line, it is negative 140 plus the slope, 14 over three, times x. Now as we can see, for most of these points, given the x value of those points, the estimate that our regression line gives is different than the actual value. And that difference between the actual and the estimate from the regression line is known as the residual. So let me write that down. So for example, the residual at that point, residual at that point, is going to be equal to, for a given x, the actual y value minus the estimated y value from the regression line for that same x. Or another way to think about it is, for that x value, when x is equal to 60, we're talking about the residual just at that point, it's going to be the actual y value minus our estimate of what the y value is from this regression line for that x value.
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Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3
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And that difference between the actual and the estimate from the regression line is known as the residual. So let me write that down. So for example, the residual at that point, residual at that point, is going to be equal to, for a given x, the actual y value minus the estimated y value from the regression line for that same x. Or another way to think about it is, for that x value, when x is equal to 60, we're talking about the residual just at that point, it's going to be the actual y value minus our estimate of what the y value is from this regression line for that x value. So pause this video and see if you can calculate this residual, and you can visually imagine it as being this right over here. Well, to actually calculate the residual, you would take our actual value, which is 125, for that x value. Remember, we're calculating the residual for a point.
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Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3
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Or another way to think about it is, for that x value, when x is equal to 60, we're talking about the residual just at that point, it's going to be the actual y value minus our estimate of what the y value is from this regression line for that x value. So pause this video and see if you can calculate this residual, and you can visually imagine it as being this right over here. Well, to actually calculate the residual, you would take our actual value, which is 125, for that x value. Remember, we're calculating the residual for a point. So it's the actual y there, minus what would be the estimated y there for that x value? Well, we could just go to this equation and say what would y hat be when x is equal to 60? Well, it's going to be equal to, let's see, we have negative 140, plus 14 over three times 60.
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Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3
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Remember, we're calculating the residual for a point. So it's the actual y there, minus what would be the estimated y there for that x value? Well, we could just go to this equation and say what would y hat be when x is equal to 60? Well, it's going to be equal to, let's see, we have negative 140, plus 14 over three times 60. Let's see, 60 divided by three is 20. 20 times 14 is 280. And so all of this is going to be 140.
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Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3
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Well, it's going to be equal to, let's see, we have negative 140, plus 14 over three times 60. Let's see, 60 divided by three is 20. 20 times 14 is 280. And so all of this is going to be 140. And so our residual for this point is gonna be 125 minus 140, which is negative 15. And residuals, indeed, can be negative. If your residual is negative, it means for that x value, your data point, your actual y value, is below the estimate.
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Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3
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And so all of this is going to be 140. And so our residual for this point is gonna be 125 minus 140, which is negative 15. And residuals, indeed, can be negative. If your residual is negative, it means for that x value, your data point, your actual y value, is below the estimate. If we were to calculate the residual here, or if we were to calculate the residual here, our actual for that x value is above our estimate. So we would get positive residuals. And as you'll see later in your statistics career, the way that we calculate these regression lines is all about minimizing the square of these residuals.
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Conditional probability explained visually (Bayes' Theorem).mp3
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One fair coin and one double-sided coin. He picks one at random, flips it, and shouts the result. Heads! Now what is the probability that he flipped the fair coin? To answer this question, we need only rewind and grow a tree. The first event, he picks one of two coins. So our tree grows two branches, leading to two equally likely outcomes, fair or unfair.
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Conditional probability explained visually (Bayes' Theorem).mp3
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Now what is the probability that he flipped the fair coin? To answer this question, we need only rewind and grow a tree. The first event, he picks one of two coins. So our tree grows two branches, leading to two equally likely outcomes, fair or unfair. The next event, he flips the coin. We grow again. If he had the fair coin, we know this flip can result in two equally likely outcomes, heads and tails.
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Conditional probability explained visually (Bayes' Theorem).mp3
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So our tree grows two branches, leading to two equally likely outcomes, fair or unfair. The next event, he flips the coin. We grow again. If he had the fair coin, we know this flip can result in two equally likely outcomes, heads and tails. While the unfair coin results in two outcomes, both heads. Our tree is finished and we see it has four leaves, representing four equally likely outcomes. The final step, new evidence.
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Conditional probability explained visually (Bayes' Theorem).mp3
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If he had the fair coin, we know this flip can result in two equally likely outcomes, heads and tails. While the unfair coin results in two outcomes, both heads. Our tree is finished and we see it has four leaves, representing four equally likely outcomes. The final step, new evidence. He says, heads! Whenever we gain evidence, we must trim our tree. We cut any branch leading to tails because we know tails did not occur.
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Conditional probability explained visually (Bayes' Theorem).mp3
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The final step, new evidence. He says, heads! Whenever we gain evidence, we must trim our tree. We cut any branch leading to tails because we know tails did not occur. And that is it. So the probability that he chose the fair coin is the one fair outcome leading to heads, divided by the three possible outcomes leading to heads, or one third. What happens if he flips again and reports, heads!
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Conditional probability explained visually (Bayes' Theorem).mp3
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We cut any branch leading to tails because we know tails did not occur. And that is it. So the probability that he chose the fair coin is the one fair outcome leading to heads, divided by the three possible outcomes leading to heads, or one third. What happens if he flips again and reports, heads! Remember, after each event, our tree grows. The fair coin leaves result in two equally likely outcomes, heads and tails. The unfair coin leaves result in two equally likely outcomes, heads and heads.
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Conditional probability explained visually (Bayes' Theorem).mp3
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What happens if he flips again and reports, heads! Remember, after each event, our tree grows. The fair coin leaves result in two equally likely outcomes, heads and tails. The unfair coin leaves result in two equally likely outcomes, heads and heads. After we hear the second, heads! We cut any branches leading to tails. Therefore, the probability the coin is fair after two heads in a row is the one fair outcome leading to heads, divided by all possible outcome leading to heads, or one fifth.
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Conditional probability explained visually (Bayes' Theorem).mp3
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The unfair coin leaves result in two equally likely outcomes, heads and heads. After we hear the second, heads! We cut any branches leading to tails. Therefore, the probability the coin is fair after two heads in a row is the one fair outcome leading to heads, divided by all possible outcome leading to heads, or one fifth. Notice our confidence in the fair coin is dropping as more heads occur, though realize it will never reach zero. No matter how many flips occur, we can never be 100% certain the coin is unfair. In fact, all conditional probability questions can be solved by growing trees.
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Conditional probability explained visually (Bayes' Theorem).mp3
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Therefore, the probability the coin is fair after two heads in a row is the one fair outcome leading to heads, divided by all possible outcome leading to heads, or one fifth. Notice our confidence in the fair coin is dropping as more heads occur, though realize it will never reach zero. No matter how many flips occur, we can never be 100% certain the coin is unfair. In fact, all conditional probability questions can be solved by growing trees. Let's do one more to be sure. Bob has three coins. Two are fair.
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Conditional probability explained visually (Bayes' Theorem).mp3
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In fact, all conditional probability questions can be solved by growing trees. Let's do one more to be sure. Bob has three coins. Two are fair. One is biased, which is weighted to land heads two thirds of the time and tails one third. He chooses a coin at random and flips it. Heads!
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Conditional probability explained visually (Bayes' Theorem).mp3
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Two are fair. One is biased, which is weighted to land heads two thirds of the time and tails one third. He chooses a coin at random and flips it. Heads! Now, what is the probability he chose the biased coin? Let's rewind and build a tree. The first event, choosing the coin, can lead to three equally likely outcomes, fair coin, fair coin, and unfair coin.
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Conditional probability explained visually (Bayes' Theorem).mp3
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Heads! Now, what is the probability he chose the biased coin? Let's rewind and build a tree. The first event, choosing the coin, can lead to three equally likely outcomes, fair coin, fair coin, and unfair coin. The next event, the coin is flipped. Each fair coin leads to two equally likely leaves, heads and tails. The biased coin leads to three equally likely leaves, two representing heads and one representing tails.
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Conditional probability explained visually (Bayes' Theorem).mp3
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The first event, choosing the coin, can lead to three equally likely outcomes, fair coin, fair coin, and unfair coin. The next event, the coin is flipped. Each fair coin leads to two equally likely leaves, heads and tails. The biased coin leads to three equally likely leaves, two representing heads and one representing tails. Now, the trick is to always make sure our tree is balanced, meaning an equal amount of leaves growing out of each branch. To do this, we simply scale up the number of branches to the least common multiple. For two and three, this is six.
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Conditional probability explained visually (Bayes' Theorem).mp3
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The biased coin leads to three equally likely leaves, two representing heads and one representing tails. Now, the trick is to always make sure our tree is balanced, meaning an equal amount of leaves growing out of each branch. To do this, we simply scale up the number of branches to the least common multiple. For two and three, this is six. And finally, we label our leaves. The fair coin now splits into six equally likely leaves, three heads and three tails. For the biased coin, we now have two tail leaves and four head leaves, and that is it.
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Conditional probability explained visually (Bayes' Theorem).mp3
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For two and three, this is six. And finally, we label our leaves. The fair coin now splits into six equally likely leaves, three heads and three tails. For the biased coin, we now have two tail leaves and four head leaves, and that is it. When Bob shows the result, heads! this new evidence allows us to trim all branches, leading to tails, since tails did not occur. So, the probability that he chose the biased coin, given heads occur?
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Conditional probability explained visually (Bayes' Theorem).mp3
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For the biased coin, we now have two tail leaves and four head leaves, and that is it. When Bob shows the result, heads! this new evidence allows us to trim all branches, leading to tails, since tails did not occur. So, the probability that he chose the biased coin, given heads occur? Well, four leaves can come from the biased coin, divided by all possible leaves. Four divided by ten, or 40%. When in doubt, it's always possible to answer conditional probability questions by Bayes' theorem.
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Conditional probability explained visually (Bayes' Theorem).mp3
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So, the probability that he chose the biased coin, given heads occur? Well, four leaves can come from the biased coin, divided by all possible leaves. Four divided by ten, or 40%. When in doubt, it's always possible to answer conditional probability questions by Bayes' theorem. It tells us the probability of event A, given some new evidence B. Though if you forgot it, no worries. You need only know how to grow stories with trimmed trees.
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Systematic random sampling AP Statistics Khan Academy.mp3
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So let's look at an example. Let's say that there is a concert that is happening, and we expect approximately 10,000 people to attend the concert. And we want to randomly sample people at the concert. Maybe we want to do a study on how do people get to the concert? How do people get to the concert? Do they drive and park? Do they ride with a friend?
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Systematic random sampling AP Statistics Khan Academy.mp3
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Maybe we want to do a study on how do people get to the concert? How do people get to the concert? Do they drive and park? Do they ride with a friend? Do they take an Uber or a cab of some kind? And so we want to find a random sample, ideally without bias, to survey people. So there's a couple of ways you could do it.
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Systematic random sampling AP Statistics Khan Academy.mp3
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Do they ride with a friend? Do they take an Uber or a cab of some kind? And so we want to find a random sample, ideally without bias, to survey people. So there's a couple of ways you could do it. You could try to do a simple random sample. And that might be a case of if you could somehow get the names of all 10,000 people and put them into a big bowl like this, and then let's say you want to sample 100 people. Let's say you want to sample approximately 100 people.
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Systematic random sampling AP Statistics Khan Academy.mp3
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So there's a couple of ways you could do it. You could try to do a simple random sample. And that might be a case of if you could somehow get the names of all 10,000 people and put them into a big bowl like this, and then let's say you want to sample 100 people. Let's say you want to sample approximately 100 people. You could just mix up all the names that may be on these little pieces of paper, 10,000 of them, and then pull them out and pull out a random sample of 100 of them. That would be a simple random sample. But you could already imagine there might be some logistic difficulties of doing this.
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Systematic random sampling AP Statistics Khan Academy.mp3
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Let's say you want to sample approximately 100 people. You could just mix up all the names that may be on these little pieces of paper, 10,000 of them, and then pull them out and pull out a random sample of 100 of them. That would be a simple random sample. But you could already imagine there might be some logistic difficulties of doing this. How are you going to get the 10,000 names? You're gonna write them on a piece of paper. That's gonna be a, you'd have to really mix some goods so it's truly random who you're picking out.
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Systematic random sampling AP Statistics Khan Academy.mp3
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But you could already imagine there might be some logistic difficulties of doing this. How are you going to get the 10,000 names? You're gonna write them on a piece of paper. That's gonna be a, you'd have to really mix some goods so it's truly random who you're picking out. So are there other ways of doing a random sample? And as you can imagine, yes, there are. And that's where systematic random sampling is useful.
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Systematic random sampling AP Statistics Khan Academy.mp3
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That's gonna be a, you'd have to really mix some goods so it's truly random who you're picking out. So are there other ways of doing a random sample? And as you can imagine, yes, there are. And that's where systematic random sampling is useful. One way to think about systematic random sampling is you're going to randomly sample a subset of the people who are maybe walking into the concert. So let's say people get to the concert and they start forming a line to get into the concert. What you wanna do in systematic random sampling is randomly pick your first person.
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Systematic random sampling AP Statistics Khan Academy.mp3
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And that's where systematic random sampling is useful. One way to think about systematic random sampling is you're going to randomly sample a subset of the people who are maybe walking into the concert. So let's say people get to the concert and they start forming a line to get into the concert. What you wanna do in systematic random sampling is randomly pick your first person. There's a bunch of ways that you could do that. Let's say you have a random number generator that'll generate a number from one to 100, and that's going to be the first person you survey if that random number generator generates a 37, then you're going to start with the 37th person in line. So you pick that first person randomly, you survey them.
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Systematic random sampling AP Statistics Khan Academy.mp3
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What you wanna do in systematic random sampling is randomly pick your first person. There's a bunch of ways that you could do that. Let's say you have a random number generator that'll generate a number from one to 100, and that's going to be the first person you survey if that random number generator generates a 37, then you're going to start with the 37th person in line. So you pick that first person randomly, you survey them. And remember, our goal is to sample about 100 people out of 10,000. So we wanna roughly sample one out of every 100 people. And so what you do there is once you have that first person that you're sampling, you then sample every 100th person after that.
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Systematic random sampling AP Statistics Khan Academy.mp3
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So you pick that first person randomly, you survey them. And remember, our goal is to sample about 100 people out of 10,000. So we wanna roughly sample one out of every 100 people. And so what you do there is once you have that first person that you're sampling, you then sample every 100th person after that. That's called sometimes the sample interval. And the reason why 100 people is because if you sample every 100th person after that, you're going to roughly get 100 people in your sample out of a total of 10,000. So this is going to be after 100, you're going to sample someone else.
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Systematic random sampling AP Statistics Khan Academy.mp3
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And so what you do there is once you have that first person that you're sampling, you then sample every 100th person after that. That's called sometimes the sample interval. And the reason why 100 people is because if you sample every 100th person after that, you're going to roughly get 100 people in your sample out of a total of 10,000. So this is going to be after 100, you're going to sample someone else. And then after another 100, you're going to sample someone else. Now, the reason why this is useful is you could say, okay, that first person was random, and then every person after that. It doesn't seem like there'd be any bias for why they would be the 100th person after that first person.
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Systematic random sampling AP Statistics Khan Academy.mp3
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So this is going to be after 100, you're going to sample someone else. And then after another 100, you're going to sample someone else. Now, the reason why this is useful is you could say, okay, that first person was random, and then every person after that. It doesn't seem like there'd be any bias for why they would be the 100th person after that first person. You don't wanna just do the first 100 people because those might be the early birds, the people who maybe disproportionately went parking or planned early or had some bias in some way. So you do wanna make sure that you're getting both the beginning, the middle, and the end of the line, which this thing helps. Now, we have to be careful.
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Systematic random sampling AP Statistics Khan Academy.mp3
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It doesn't seem like there'd be any bias for why they would be the 100th person after that first person. You don't wanna just do the first 100 people because those might be the early birds, the people who maybe disproportionately went parking or planned early or had some bias in some way. So you do wanna make sure that you're getting both the beginning, the middle, and the end of the line, which this thing helps. Now, we have to be careful. Even systematic random sampling is not foolproof. There's a situation where inadvertently, even this system has bias. Let's say that this is the arena.
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Systematic random sampling AP Statistics Khan Academy.mp3
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Now, we have to be careful. Even systematic random sampling is not foolproof. There's a situation where inadvertently, even this system has bias. Let's say that this is the arena. This is a top view of the arena right over here. And this is the line of people coming in. And this is where you are standing, and you are counting every 100th person.
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Systematic random sampling AP Statistics Khan Academy.mp3
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Let's say that this is the arena. This is a top view of the arena right over here. And this is the line of people coming in. And this is where you are standing, and you are counting every 100th person. But maybe, and let's say there's a tree right over here, and maybe there's a road. I'm making this quite elaborate. So maybe there is a road right over here.
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Systematic random sampling AP Statistics Khan Academy.mp3
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And this is where you are standing, and you are counting every 100th person. But maybe, and let's say there's a tree right over here, and maybe there's a road. I'm making this quite elaborate. So maybe there is a road right over here. And a lot of people, maybe all of the people who are walking or taking a cab are coming from this direction. And maybe all of the people from the parking lot are coming from this direction. And maybe you have a police officer right over here who is doing crowd control, who lets 50 of these people in, followed by 50 of these people in.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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The first time you're exposed to permutations and combinations, it takes a little bit to get your brain around it. So I think it never hurts to do as many examples. But each incremental example, I'm gonna go, I'm gonna review what we've done before, but hopefully go a little bit further. So let's just take another example. And this is in the same vein. In videos after this, I'll start using other examples other than just people sitting in chairs. But let's stick with it for now.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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So let's just take another example. And this is in the same vein. In videos after this, I'll start using other examples other than just people sitting in chairs. But let's stick with it for now. So let's say we have six people again. So person A, B, C, D, E, and F. So we have six people. And now let's put them into four chairs.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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But let's stick with it for now. So let's say we have six people again. So person A, B, C, D, E, and F. So we have six people. And now let's put them into four chairs. We can go through this fairly quickly. One, two, three, four chairs. And we've seen this show multiple times.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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And now let's put them into four chairs. We can go through this fairly quickly. One, two, three, four chairs. And we've seen this show multiple times. How many ways, how many permutations are there of putting these six people into four chairs? Well, the first chair, if we seat them in order, we might as well, we could say, well, there'd be six possibilities here. Now for each of those six possibilities, there'd be five possibilities of who we put here, because one person's already sitting down.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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And we've seen this show multiple times. How many ways, how many permutations are there of putting these six people into four chairs? Well, the first chair, if we seat them in order, we might as well, we could say, well, there'd be six possibilities here. Now for each of those six possibilities, there'd be five possibilities of who we put here, because one person's already sitting down. Now for each of these 30 possibilities of seating these first two people, there'd be four possibilities of who we put in chair number three. And then for each of these, what is this, 120 possibilities, there would be three possibilities of who we put in chair four. And so this six times four times, six times five times four times three is the number of permutations.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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Now for each of those six possibilities, there'd be five possibilities of who we put here, because one person's already sitting down. Now for each of these 30 possibilities of seating these first two people, there'd be four possibilities of who we put in chair number three. And then for each of these, what is this, 120 possibilities, there would be three possibilities of who we put in chair four. And so this six times four times, six times five times four times three is the number of permutations. And we've seen in one of the early videos on permutations that, or when we talk about the permutation formula, one way to write this, if we wanted to write it in terms of factorial, we could write this as six factorial, six factorial, which is going to be equal to six times five times four times three times two times one. But we want to get rid of the two times one. So we're gonna divide that.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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And so this six times four times, six times five times four times three is the number of permutations. And we've seen in one of the early videos on permutations that, or when we talk about the permutation formula, one way to write this, if we wanted to write it in terms of factorial, we could write this as six factorial, six factorial, which is going to be equal to six times five times four times three times two times one. But we want to get rid of the two times one. So we're gonna divide that. We're gonna divide that. Now what's two times one? Well, two times one is two factorial, and where did we get that?
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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So we're gonna divide that. We're gonna divide that. Now what's two times one? Well, two times one is two factorial, and where did we get that? Well, we wanted the first four, the first four factors of six factorial. So if you want, and that's where the four came from. We wanted the first four factors, and so the way we got two is we said six minus four.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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Well, two times one is two factorial, and where did we get that? Well, we wanted the first four, the first four factors of six factorial. So if you want, and that's where the four came from. We wanted the first four factors, and so the way we got two is we said six minus four. Six minus four, that's going to get us what we want to get, that's going to give us the number that we want to get rid of. So we wanted to get rid of two, or the factors we want to get rid of. So that's going to give us two factorial.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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We wanted the first four factors, and so the way we got two is we said six minus four. Six minus four, that's going to get us what we want to get, that's going to give us the number that we want to get rid of. So we wanted to get rid of two, or the factors we want to get rid of. So that's going to give us two factorial. So if we use six minus four factorial, then that's going to give us two factorial, which is two times one, and then these cancel out, and we are all set. And so this is one way, this is, you know, I put in the particular numbers here, but this is a review of the permutations formula, where people say, hey, if I'm saying n, if I'm taking n things, and I want to figure out how many permutations are there of putting them into, let's say, k spots, it's going to be equal to n factorial over n minus k factorial. That's exactly what we did over here, where six is n, and k, or four, is k. Four is k. Actually, let me color code the whole thing so that we see the parallel.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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So that's going to give us two factorial. So if we use six minus four factorial, then that's going to give us two factorial, which is two times one, and then these cancel out, and we are all set. And so this is one way, this is, you know, I put in the particular numbers here, but this is a review of the permutations formula, where people say, hey, if I'm saying n, if I'm taking n things, and I want to figure out how many permutations are there of putting them into, let's say, k spots, it's going to be equal to n factorial over n minus k factorial. That's exactly what we did over here, where six is n, and k, or four, is k. Four is k. Actually, let me color code the whole thing so that we see the parallel. Now, all of that is review, but then we went into the world of combinations, and in the world of combinations, we said, okay, permutations make a difference between who's sitting in what chair. So for example, in the permutations world, and this is all review, we've covered this in the first combinations video, in the permutations world, A, B, C, D, and D, A, B, C, these would be two different permutations. It's being counted in whatever number this is.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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That's exactly what we did over here, where six is n, and k, or four, is k. Four is k. Actually, let me color code the whole thing so that we see the parallel. Now, all of that is review, but then we went into the world of combinations, and in the world of combinations, we said, okay, permutations make a difference between who's sitting in what chair. So for example, in the permutations world, and this is all review, we've covered this in the first combinations video, in the permutations world, A, B, C, D, and D, A, B, C, these would be two different permutations. It's being counted in whatever number this is. This is what? This is 30 times 12. This is equal to 360.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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It's being counted in whatever number this is. This is what? This is 30 times 12. This is equal to 360. So this is, each of these, this is one permutation. This is another permutation, and if we keep doing it, we would count up to 360. But we learned in combinations, when we're thinking about combinations, let me write combinations.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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This is equal to 360. So this is, each of these, this is one permutation. This is another permutation, and if we keep doing it, we would count up to 360. But we learned in combinations, when we're thinking about combinations, let me write combinations. So if we're saying N choose K, or how many combinations are there, if we take K things, and we just want to figure out how many combinations, sorry, if we start with N, if we have a pool of N things, and we want to say how many combinations of K things are there, then we would count these as the same combination. So what we really want to do is we want to take the number of permutations there are, we want to take the number of permutations there are, which is equal to N factorial over N minus K factorial, over N minus K factorial, and we want to divide by the number of ways that you could arrange four people. Once again, and this takes, I remember the first time I learned it, it took my brain a little while, so if it's taking a little while to think about it, not a big deal.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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But we learned in combinations, when we're thinking about combinations, let me write combinations. So if we're saying N choose K, or how many combinations are there, if we take K things, and we just want to figure out how many combinations, sorry, if we start with N, if we have a pool of N things, and we want to say how many combinations of K things are there, then we would count these as the same combination. So what we really want to do is we want to take the number of permutations there are, we want to take the number of permutations there are, which is equal to N factorial over N minus K factorial, over N minus K factorial, and we want to divide by the number of ways that you could arrange four people. Once again, and this takes, I remember the first time I learned it, it took my brain a little while, so if it's taking a little while to think about it, not a big deal. It can be confusing at first, but it'll hopefully, if you keep thinking about it, hopefully you will see clarity at some moment. But what we want to do is we want to divide by all of the ways that you could arrange four things, because once again, in the permutations, it's counting all of the different arrangements of four things, but we don't want to count all of those different arrangements of four things. We want to just say, well, they're all one combination.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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Once again, and this takes, I remember the first time I learned it, it took my brain a little while, so if it's taking a little while to think about it, not a big deal. It can be confusing at first, but it'll hopefully, if you keep thinking about it, hopefully you will see clarity at some moment. But what we want to do is we want to divide by all of the ways that you could arrange four things, because once again, in the permutations, it's counting all of the different arrangements of four things, but we don't want to count all of those different arrangements of four things. We want to just say, well, they're all one combination. So we want to divide by the number of ways to arrange four things, or the number of ways to arrange K things. So let me write this down. So what is the number of ways, number of ways to arrange K things, K things in K spots?
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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We want to just say, well, they're all one combination. So we want to divide by the number of ways to arrange four things, or the number of ways to arrange K things. So let me write this down. So what is the number of ways, number of ways to arrange K things, K things in K spots? And I encourage you to pause the video, because this is actually a review from the first permutation video. Well, if you have K spots, let me do it. So this is the first spot, the second spot, third spot, and then you're going to go all the way to the Kth spot.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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So what is the number of ways, number of ways to arrange K things, K things in K spots? And I encourage you to pause the video, because this is actually a review from the first permutation video. Well, if you have K spots, let me do it. So this is the first spot, the second spot, third spot, and then you're going to go all the way to the Kth spot. Well, for the first spot, there could be K possibilities. There's K things that could take the first spot. Now, for each of those K possibilities, how many things could be in the second spot?
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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So this is the first spot, the second spot, third spot, and then you're going to go all the way to the Kth spot. Well, for the first spot, there could be K possibilities. There's K things that could take the first spot. Now, for each of those K possibilities, how many things could be in the second spot? Well, it's going to be K minus one, because you already put something in the first spot, and then over here, what's it going to be? K minus two, all the way to the last spot. There's only one thing that could be put in the last spot.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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Now, for each of those K possibilities, how many things could be in the second spot? Well, it's going to be K minus one, because you already put something in the first spot, and then over here, what's it going to be? K minus two, all the way to the last spot. There's only one thing that could be put in the last spot. So what is this thing here? K times K minus one times K minus two times K minus three, all the way down to one. Well, this is just equal to K factorial.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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There's only one thing that could be put in the last spot. So what is this thing here? K times K minus one times K minus two times K minus three, all the way down to one. Well, this is just equal to K factorial. The number of ways to arrange K things in K spots, K factorial. The number of ways to arrange four things in four spots, that's four factorial. The number of ways to arrange three things in three spots, it's three factorial.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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Well, this is just equal to K factorial. The number of ways to arrange K things in K spots, K factorial. The number of ways to arrange four things in four spots, that's four factorial. The number of ways to arrange three things in three spots, it's three factorial. So we could just divide this. We could just divide this by K factorial, and so this would get us, this would get us N factorial divided by K factorial, K factorial times, times N minus K factorial. N minus K, N minus K, and I'll put the factorial right over there.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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The number of ways to arrange three things in three spots, it's three factorial. So we could just divide this. We could just divide this by K factorial, and so this would get us, this would get us N factorial divided by K factorial, K factorial times, times N minus K factorial. N minus K, N minus K, and I'll put the factorial right over there. And this right over here is the formula, this right over here is the formula for combinations. Sometimes this is also called the binomial coefficient. Some people will call this N choose K. They'll also write it like this.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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N minus K, N minus K, and I'll put the factorial right over there. And this right over here is the formula, this right over here is the formula for combinations. Sometimes this is also called the binomial coefficient. Some people will call this N choose K. They'll also write it like this. N choose K, especially when they're thinking in terms of binomial coefficients. But I got into kind of an abstract tangent here when I started getting into the general formula, but let's go back to our example. So in our example, we saw there was 360 ways of seating six people into four chairs.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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Some people will call this N choose K. They'll also write it like this. N choose K, especially when they're thinking in terms of binomial coefficients. But I got into kind of an abstract tangent here when I started getting into the general formula, but let's go back to our example. So in our example, we saw there was 360 ways of seating six people into four chairs. But what if we didn't care about who's sitting in which chairs, and we just want to say, how many ways are there to choose four people from a pool of six? Well, that would be, that would be how many ways are there, so that would be six, how many combinations, if I'm starting with a pool of six, how many combinations are there, how many combinations are there for selecting four? Or another way of thinking about it is, how many ways are there to, from a pool of six items, people in this example, how many ways are there to choose four of them?
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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So in our example, we saw there was 360 ways of seating six people into four chairs. But what if we didn't care about who's sitting in which chairs, and we just want to say, how many ways are there to choose four people from a pool of six? Well, that would be, that would be how many ways are there, so that would be six, how many combinations, if I'm starting with a pool of six, how many combinations are there, how many combinations are there for selecting four? Or another way of thinking about it is, how many ways are there to, from a pool of six items, people in this example, how many ways are there to choose four of them? And that is going to be, you know, we could do it, you know, well, I'll apply the formula first, and then I'll reason through it. And like I always say, I don't, I'm not a huge fan of the formula. Every time I, you know, I revisit it after a few years, I actually just rethink about it as opposed to memorizing it, because memorizing is a good way to not understand what's actually going on.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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Or another way of thinking about it is, how many ways are there to, from a pool of six items, people in this example, how many ways are there to choose four of them? And that is going to be, you know, we could do it, you know, well, I'll apply the formula first, and then I'll reason through it. And like I always say, I don't, I'm not a huge fan of the formula. Every time I, you know, I revisit it after a few years, I actually just rethink about it as opposed to memorizing it, because memorizing is a good way to not understand what's actually going on. But if we just applied the formula here, but I really want you to understand what's happening with the formula, it would be six factorial over four factorial, over four factorial, times six minus four factorial. Six, whoops, let me actually, let me just, so this is six minus four factorial, so this part right over here, six minus four, actually, let me write it out, because I know this can be a little bit confusing the first time you see it. So six minus four factorial, factorial, which is equal to, which is equal to six factorial over four factorial, over four factorial, times, this thing right over here is two factorial, times two factorial, which is going to be equal to, we could just write out the factorial, six times five times four times three times two times one, over four, four times three times two times one, times, times two times one, and of course, that's going to cancel with that, and then the one really doesn't change the value, so let me get rid of this one here, and then let's see, this three can cancel with this three, this four could cancel with this four, and then it's six divided by two is going to be three, and so we are just left with three times five, so we are left with, we are left with, there's 15 combinations.
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Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3
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Every time I, you know, I revisit it after a few years, I actually just rethink about it as opposed to memorizing it, because memorizing is a good way to not understand what's actually going on. But if we just applied the formula here, but I really want you to understand what's happening with the formula, it would be six factorial over four factorial, over four factorial, times six minus four factorial. Six, whoops, let me actually, let me just, so this is six minus four factorial, so this part right over here, six minus four, actually, let me write it out, because I know this can be a little bit confusing the first time you see it. So six minus four factorial, factorial, which is equal to, which is equal to six factorial over four factorial, over four factorial, times, this thing right over here is two factorial, times two factorial, which is going to be equal to, we could just write out the factorial, six times five times four times three times two times one, over four, four times three times two times one, times, times two times one, and of course, that's going to cancel with that, and then the one really doesn't change the value, so let me get rid of this one here, and then let's see, this three can cancel with this three, this four could cancel with this four, and then it's six divided by two is going to be three, and so we are just left with three times five, so we are left with, we are left with, there's 15 combinations. There's 360 permutations for putting six people into four chairs, but there's only 15 combinations, because we're no longer counting all of the different arrangements for the same four people in the four chairs. We're saying, hey, if it's the same four people, that is now one combination, and you can see how many ways are there to arrange four people into four chairs? Well, that's the four factorial part right over here, the four factorial part right over here, which is four times three times two times one, which is 24, so we essentially just took the 360 divided by 24 to get 15, but once again, I don't want to, I don't think I can stress this enough.
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Example Different ways to pick officers Precalculus Khan Academy.mp3
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A club of nine people wants to choose a board of three officers, president, vice president, and secretary. Assuming the officers are chosen at random, what is the probability that the officers are Marsha for president, Sabitha for vice president, and Robert for secretary? So to think about the probability of Marsha, so let me write this president, president is equal to Marsha, or vice president is equal to Sabitha, and secretary is equal to Robert. This is going to be, this right here is one possible outcome, one specific outcome, so it's one outcome out of the total number of outcomes over the total number of possibilities. Now what is the total number of possibilities? Well to think about that, let's just think about the three positions. You have president, you have vice president, and you have secretary.
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Example Different ways to pick officers Precalculus Khan Academy.mp3
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This is going to be, this right here is one possible outcome, one specific outcome, so it's one outcome out of the total number of outcomes over the total number of possibilities. Now what is the total number of possibilities? Well to think about that, let's just think about the three positions. You have president, you have vice president, and you have secretary. Now let's just assume that we're going to fill the slot of president first. We don't have to do president first, but we're just going to pick here. So if we're just picking president first, we haven't assigned anyone to any officers just yet, so we have nine people to choose from.
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Example Different ways to pick officers Precalculus Khan Academy.mp3
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You have president, you have vice president, and you have secretary. Now let's just assume that we're going to fill the slot of president first. We don't have to do president first, but we're just going to pick here. So if we're just picking president first, we haven't assigned anyone to any officers just yet, so we have nine people to choose from. So there are nine possibilities here. Now when we go to selecting our vice president, we would have already assigned one person to the president. So we only have eight people to pick from.
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Example Different ways to pick officers Precalculus Khan Academy.mp3
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So if we're just picking president first, we haven't assigned anyone to any officers just yet, so we have nine people to choose from. So there are nine possibilities here. Now when we go to selecting our vice president, we would have already assigned one person to the president. So we only have eight people to pick from. And when we assign our secretary, we would have already assigned our president and vice president, so we're only going to have seven people to pick from. So the total permutations here, or the total number of possibilities, or the total number of ways to pick president, vice president, and secretary from nine people is going to be 9 times 8 times 7, which is, let's see, 9 times 8 is 72, 72 times 7, 2 times 7 is 14, 7 times 7 is 49, plus 1 is 50. So there's 504 possibilities.
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Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3
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Before applying to law school in the US, students need to take an exam called the LSAT. Before applying to medical school, students need to take an exam called the MCAT. Here are some summary statistics for each exam. So the LSAT, the mean score is 151 with a standard deviation of 10. And the MCAT, the mean score is 25.1 with a standard deviation of 6.4. Juwan took both exams. He scored 172 on the LSAT and 37 on the MCAT.
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Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3
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So the LSAT, the mean score is 151 with a standard deviation of 10. And the MCAT, the mean score is 25.1 with a standard deviation of 6.4. Juwan took both exams. He scored 172 on the LSAT and 37 on the MCAT. Which exam did he do relatively better on? So pause this video and see if you can figure it out. So the way I would think about it is, you can't just look at the absolute score because they are on different scales and they have different distributions.
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Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3
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He scored 172 on the LSAT and 37 on the MCAT. Which exam did he do relatively better on? So pause this video and see if you can figure it out. So the way I would think about it is, you can't just look at the absolute score because they are on different scales and they have different distributions. But we can use this information, if we assume it's a normal distribution or relatively close to a normal distribution with a mean centered at this mean, we can think about, well, how many standard deviations from the mean did he score in each of these situations? So in both cases, he scored above the mean, but how many standard deviations above the mean? So let's see if we can figure that out.
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Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3
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So the way I would think about it is, you can't just look at the absolute score because they are on different scales and they have different distributions. But we can use this information, if we assume it's a normal distribution or relatively close to a normal distribution with a mean centered at this mean, we can think about, well, how many standard deviations from the mean did he score in each of these situations? So in both cases, he scored above the mean, but how many standard deviations above the mean? So let's see if we can figure that out. So on the LSAT, let's see, let me write this down. On the LSAT, he scored 172. So how many standard deviations is that going to be?
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Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3
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So let's see if we can figure that out. So on the LSAT, let's see, let me write this down. On the LSAT, he scored 172. So how many standard deviations is that going to be? Well, let's take 172, his score, minus the mean. So this is the absolute number that he scored above the mean. And now let's divide that by the standard deviation.
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Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3
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So how many standard deviations is that going to be? Well, let's take 172, his score, minus the mean. So this is the absolute number that he scored above the mean. And now let's divide that by the standard deviation. So on the LSAT, this is what? This is going to be 21 divided by 10. So this is 2.1 standard deviations.
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Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3
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And now let's divide that by the standard deviation. So on the LSAT, this is what? This is going to be 21 divided by 10. So this is 2.1 standard deviations. Deviations above the mean, above the mean. You could view this as a z-score. It's a z-score of 2.1.
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Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3
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So this is 2.1 standard deviations. Deviations above the mean, above the mean. You could view this as a z-score. It's a z-score of 2.1. We are 2.1 above the mean in this situation. Now let's think about how he did on the MCAT. On the MCAT, he scored a 37.
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Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3
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It's a z-score of 2.1. We are 2.1 above the mean in this situation. Now let's think about how he did on the MCAT. On the MCAT, he scored a 37. He scored a 37. The mean is a 25.1. And there is a standard deviation of 6.4.
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Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3
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On the MCAT, he scored a 37. He scored a 37. The mean is a 25.1. And there is a standard deviation of 6.4. So let's see, 37.1 minus 25 would be 12, but now it's gonna be 11.9. 11.9 divided by 6.4. So without even looking at this, this is going to be approximately, well, this is gonna be a little bit less than two.
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Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3
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And there is a standard deviation of 6.4. So let's see, 37.1 minus 25 would be 12, but now it's gonna be 11.9. 11.9 divided by 6.4. So without even looking at this, this is going to be approximately, well, this is gonna be a little bit less than two. This is going to be less than two. So based on this information, and we could figure out the exact number here. In fact, let me get my calculator out.
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Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3
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So without even looking at this, this is going to be approximately, well, this is gonna be a little bit less than two. This is going to be less than two. So based on this information, and we could figure out the exact number here. In fact, let me get my calculator out. So you get the calculator. So if we do 11.9 divided by 6.4, that's gonna get us to 1 point, I'll just say approximately 1.86. So approximately 1.86.
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Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3
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In fact, let me get my calculator out. So you get the calculator. So if we do 11.9 divided by 6.4, that's gonna get us to 1 point, I'll just say approximately 1.86. So approximately 1.86. So relatively speaking, he did slightly better on the LSAT. He did more standard deviations, although this is close. I would say they're comparable.
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Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3
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So approximately 1.86. So relatively speaking, he did slightly better on the LSAT. He did more standard deviations, although this is close. I would say they're comparable. He did roughly two standard deviations if we were around to the nearest standard deviation, but if you wanted to get precise, he did a little bit better, relatively speaking, on the LSAT. He did 2.1 standard deviations here, while over here he did 1.86 or 1.9 standard deviations. But in everyday language, you would probably say, well, this is comparable.
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Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3
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How many different possibilities are there? And to make that a little bit tangible, let's have an example with, say, a sofa. My sofa can seat exactly three people. I have seat number one on the left of the sofa, seat number two in the middle of the sofa, and seat number three on the right of the sofa. And let's say we're going to have three people who are going to sit in these three seats, person A, person B, and person C. How many different ways can these three people sit in these three seats? Pause this video and see if you can figure it out on your own. Well, there are several ways to approach this.
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Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3
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I have seat number one on the left of the sofa, seat number two in the middle of the sofa, and seat number three on the right of the sofa. And let's say we're going to have three people who are going to sit in these three seats, person A, person B, and person C. How many different ways can these three people sit in these three seats? Pause this video and see if you can figure it out on your own. Well, there are several ways to approach this. One way is to just try to think through all of the possibilities. You could do it systematically. You could say, all right, if I have person A in seat number one, then I could have person B in seat number two and person C in seat number three.
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Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3
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Well, there are several ways to approach this. One way is to just try to think through all of the possibilities. You could do it systematically. You could say, all right, if I have person A in seat number one, then I could have person B in seat number two and person C in seat number three. And I could think of another situation. If I have person A in seat number one, I could then swap B and C, so it could look like that. And that's all of the situations, all of the permutations where I have A in seat number one.
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Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3
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You could say, all right, if I have person A in seat number one, then I could have person B in seat number two and person C in seat number three. And I could think of another situation. If I have person A in seat number one, I could then swap B and C, so it could look like that. And that's all of the situations, all of the permutations where I have A in seat number one. So now let's put someone else in seat number one. So now let's put B in seat number one, and I could put A in the middle and C on the right. Or I could put B in seat number one and then swap A and C. So C and then A.
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Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3
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And that's all of the situations, all of the permutations where I have A in seat number one. So now let's put someone else in seat number one. So now let's put B in seat number one, and I could put A in the middle and C on the right. Or I could put B in seat number one and then swap A and C. So C and then A. And then if I put C in seat number one, well, I could put A in the middle and B on the right. Or with C in seat number one, I could put B in the middle and A on the right. And these are actually all of the permutations.
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Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3
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Or I could put B in seat number one and then swap A and C. So C and then A. And then if I put C in seat number one, well, I could put A in the middle and B on the right. Or with C in seat number one, I could put B in the middle and A on the right. And these are actually all of the permutations. And you can see that there are one, two, three, four, five, six. Now this wasn't too bad, and in general, if you're thinking about permutations of six things or three things in three spaces, you can do it by hand. But it could get very complicated if I said, hey, I have 100 seats and I have 100 people that are going to sit in them.
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Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3
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And these are actually all of the permutations. And you can see that there are one, two, three, four, five, six. Now this wasn't too bad, and in general, if you're thinking about permutations of six things or three things in three spaces, you can do it by hand. But it could get very complicated if I said, hey, I have 100 seats and I have 100 people that are going to sit in them. How do I figure it out mathematically? Well, the way that you would do it, and this is going to be a technique that you can use for really any number of people in any number of seats, is to really just build off of what we just did here. What we did here is we started with seat number one.
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