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There's only one way to get that. It's the remaining cards that's going to give all of the different combinations of having four 1's. So this will be a count of all of the different combinations. Because all of the different extra stuff that you have will be all of the different hands. Now, we know the total number of hands with four 1's. It's this number. And now we can divide it by the total number of possible hands. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Because all of the different extra stuff that you have will be all of the different hands. Now, we know the total number of hands with four 1's. It's this number. And now we can divide it by the total number of possible hands. And I didn't multiply them out on purpose so that we can cancel things out. So let's do that. Let's take this and divide by that. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And now we can divide it by the total number of possible hands. And I didn't multiply them out on purpose so that we can cancel things out. So let's do that. Let's take this and divide by that. So let me just copy and paste it. Let me take that. Let me copy it and let me paste it. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Let's take this and divide by that. So let me just copy and paste it. Let me take that. Let me copy it and let me paste it. Let's take that and let's divide it by that. But dividing by a fraction is the same thing as multiplying by the reciprocal. So let's just multiply by the reciprocal. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Let me copy it and let me paste it. Let's take that and let's divide it by that. But dividing by a fraction is the same thing as multiplying by the reciprocal. So let's just multiply by the reciprocal. So let's multiply. So this is the denominator. Let's make this the numerator. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So let's just multiply by the reciprocal. So let's multiply. So this is the denominator. Let's make this the numerator. So let me copy it and then let me paste it. So that's the numerator. And then that's the denominator up there. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Let's make this the numerator. So let me copy it and then let me paste it. So that's the numerator. And then that's the denominator up there. Because we're dividing by that expression. So let me put that there. Let me get the Select tool. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And then that's the denominator up there. Because we're dividing by that expression. So let me put that there. Let me get the Select tool. And then let me make sure I'm selecting all of the numbers. Let me copy it and then let me paste that. It's a little messy with those lines there, but I think this will suit our purposes. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Let me get the Select tool. And then let me make sure I'm selecting all of the numbers. Let me copy it and then let me paste that. It's a little messy with those lines there, but I think this will suit our purposes. So when we're multiplying by this, we're essentially dividing by this expression up here. Now, this we can simplify pretty easily. We have a, well actually I forgot to do, this should be 9 factorial. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
It's a little messy with those lines there, but I think this will suit our purposes. So when we're multiplying by this, we're essentially dividing by this expression up here. Now, this we can simplify pretty easily. We have a, well actually I forgot to do, this should be 9 factorial. This should be 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1. So let me put that in both places. Actually, let me just clear that both places. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
We have a, well actually I forgot to do, this should be 9 factorial. This should be 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1. So let me put that in both places. Actually, let me just clear that both places. I'm sorry if that confused you when I wrote it earlier. This would be 9 factorial 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1. Let me copy and paste that now. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Actually, let me just clear that both places. I'm sorry if that confused you when I wrote it earlier. This would be 9 factorial 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1. Let me copy and paste that now. Copy and then you paste it. That's that right there. And then we have this in the numerator. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Let me copy and paste that now. Copy and then you paste it. That's that right there. And then we have this in the numerator. We have 5 times 4 times 3 times 1 in the denominator. So this will cancel out with that part right over there. And then we have 32 times 31 times 30 times 29 times 28. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And then we have this in the numerator. We have 5 times 4 times 3 times 1 in the denominator. So this will cancel out with that part right over there. And then we have 32 times 31 times 30 times 29 times 28. That is going to cancel with that. That and that cancels out. So what we're left with is just this part over here. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And then we have 32 times 31 times 30 times 29 times 28. That is going to cancel with that. That and that cancels out. So what we're left with is just this part over here. Let me rewrite it. So we're left with 9 times 8 times 7 times 6 over, and this will just be an exercise in simplifying this expression, 36 times 35 times 34 times 33. And let's see. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So what we're left with is just this part over here. Let me rewrite it. So we're left with 9 times 8 times 7 times 6 over, and this will just be an exercise in simplifying this expression, 36 times 35 times 34 times 33. And let's see. If we divide the numerator and denominator by 9, that becomes a 1. This becomes a 4. You can divide the numerator and denominator by 4. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And let's see. If we divide the numerator and denominator by 9, that becomes a 1. This becomes a 4. You can divide the numerator and denominator by 4. This becomes a 2. This becomes a 1. You divide numerator and denominator by 7. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
You can divide the numerator and denominator by 4. This becomes a 2. This becomes a 1. You divide numerator and denominator by 7. This becomes a 1. This becomes a 5. You can divide both by 2 again. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
You divide numerator and denominator by 7. This becomes a 1. This becomes a 5. You can divide both by 2 again. And then this becomes a 1. This becomes a 17. And you could divide this and this by 3. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
You can divide both by 2 again. And then this becomes a 1. This becomes a 17. And you could divide this and this by 3. This becomes a 2. And then this becomes an 11. So we're left with the probability of having all 4 1's in my hand of 9 that I'm selecting from 36 unique cards is equal to, and the numerator I'm just left with this 2, times 1 times 1 times 1. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And you could divide this and this by 3. This becomes a 2. And then this becomes an 11. So we're left with the probability of having all 4 1's in my hand of 9 that I'm selecting from 36 unique cards is equal to, and the numerator I'm just left with this 2, times 1 times 1 times 1. So it's equal to 2 over 5 times 17 times 11. And that is, so drum roll, this was kind of an involved problem, 5 times 17 times 11 is equal to 935. So it's equal to 2 over 935. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
She recorded the height in centimeters of each customer and the frame size in centimeters of the bicycle that customer rented. After plotting her results, Vera noticed that the relationship between the two variables was fairly linear, so she used the data to calculate the following least squares regression equation for predicting bicycle frame size from the height of the customer, and this is the equation. So before I even look at this question, let's just think about what she did. So she had a bunch of customers, and she recorded, given the height of the customer, what size frame that person rented, and so she might have had something like this, where in the horizontal axis, you have height measured in centimeters, and in the vertical axis, you have frame size that's also measured in centimeters, and so there might have been someone who measures 100 centimeters in height who gets a 25-centimeter frame. I don't know if that's reasonable or not for you bicycle experts, but let's just go with it, and so she would have plotted it there. Maybe there was another person of 100 centimeters in height who got a frame that was slightly larger, and she plotted it there, and then she did a least squares regression, and a least squares regression is trying to fit a line to this data. Oftentimes, you would use a spreadsheet or you use a computer, and that line is trying to minimize the square of the distance between these points, and so the least squares regression, maybe it would look something like this, and this is just a rough estimate of it. | Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
So she had a bunch of customers, and she recorded, given the height of the customer, what size frame that person rented, and so she might have had something like this, where in the horizontal axis, you have height measured in centimeters, and in the vertical axis, you have frame size that's also measured in centimeters, and so there might have been someone who measures 100 centimeters in height who gets a 25-centimeter frame. I don't know if that's reasonable or not for you bicycle experts, but let's just go with it, and so she would have plotted it there. Maybe there was another person of 100 centimeters in height who got a frame that was slightly larger, and she plotted it there, and then she did a least squares regression, and a least squares regression is trying to fit a line to this data. Oftentimes, you would use a spreadsheet or you use a computer, and that line is trying to minimize the square of the distance between these points, and so the least squares regression, maybe it would look something like this, and this is just a rough estimate of it. It might look something, actually, let me get my ruler tool. It might look something like, it might look something like this. So let me plot it. | Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
Oftentimes, you would use a spreadsheet or you use a computer, and that line is trying to minimize the square of the distance between these points, and so the least squares regression, maybe it would look something like this, and this is just a rough estimate of it. It might look something, actually, let me get my ruler tool. It might look something like, it might look something like this. So let me plot it. So this, that would be the line, so our regression line, y-hat, is equal to 1 3rd plus 1 3rd x, and so you could view this as a way of predicting or either modeling the relationship or predicting that, hey, if I get a new person, I could take their height and put it as an x and figure out what frame size they're likely to rent, but they ask us, what is the residual of a customer with a frame, with a height of 155 centimeters who rents a bike with a 51-centimeter frame? So how do we think about this? Well, the residual is going to be the difference between what they actually produce and what the line, what our regression line would have predicted, so we could say residual, let me write it this way, residual is going to be actual, actual minus predicted. | Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
So let me plot it. So this, that would be the line, so our regression line, y-hat, is equal to 1 3rd plus 1 3rd x, and so you could view this as a way of predicting or either modeling the relationship or predicting that, hey, if I get a new person, I could take their height and put it as an x and figure out what frame size they're likely to rent, but they ask us, what is the residual of a customer with a frame, with a height of 155 centimeters who rents a bike with a 51-centimeter frame? So how do we think about this? Well, the residual is going to be the difference between what they actually produce and what the line, what our regression line would have predicted, so we could say residual, let me write it this way, residual is going to be actual, actual minus predicted. So if predicted is larger than actual, this is actually going to be a negative number. If predicted is smaller than actual, this is going to be a positive number. Well, we know the actual, they tell us that, they tell us that they rent, it's a, the 155-centimeter person rents a bike with a 51-centimeter frame, so this is 51 centimeters, but what is the predicted? | Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
Well, the residual is going to be the difference between what they actually produce and what the line, what our regression line would have predicted, so we could say residual, let me write it this way, residual is going to be actual, actual minus predicted. So if predicted is larger than actual, this is actually going to be a negative number. If predicted is smaller than actual, this is going to be a positive number. Well, we know the actual, they tell us that, they tell us that they rent, it's a, the 155-centimeter person rents a bike with a 51-centimeter frame, so this is 51 centimeters, but what is the predicted? Well, that's where we can use our regression equation that Vera came up with. The predicted, I'll do that in orange, the predicted is going to be equal to 1 3rd plus 1 3rd times a person's height. Their height is 155. | Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
Well, we know the actual, they tell us that, they tell us that they rent, it's a, the 155-centimeter person rents a bike with a 51-centimeter frame, so this is 51 centimeters, but what is the predicted? Well, that's where we can use our regression equation that Vera came up with. The predicted, I'll do that in orange, the predicted is going to be equal to 1 3rd plus 1 3rd times a person's height. Their height is 155. That's the predicted. Y hat is what our linear regression predicts, our line predicts, so what is this going to be? This is going to be equal to 1 3rd plus 155 over three, which is equal to 156 over three, which comes out nicely to 52. | Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
Their height is 155. That's the predicted. Y hat is what our linear regression predicts, our line predicts, so what is this going to be? This is going to be equal to 1 3rd plus 155 over three, which is equal to 156 over three, which comes out nicely to 52. So the predicted on our line is 52, and so here, so this person is 155, we can plot them right over here, 155. They're coming in slightly below the line. So they're coming in slightly below the line right there, and that distance, which is, and we can see that they are below the line, so that distance is going to be, or in this case, the residual is going to be negative, so this is going to be negative one. | Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
This is going to be equal to 1 3rd plus 155 over three, which is equal to 156 over three, which comes out nicely to 52. So the predicted on our line is 52, and so here, so this person is 155, we can plot them right over here, 155. They're coming in slightly below the line. So they're coming in slightly below the line right there, and that distance, which is, and we can see that they are below the line, so that distance is going to be, or in this case, the residual is going to be negative, so this is going to be negative one. And so if we were to zoom in right over here, you can't see it that well, but let me draw, so if we zoom in, let's say we were to zoom in the line, and it looks like this, and our data point is right, our data point is right over here. We know we're below the line, and this is going to be a negative residual, and the magnitude of that residual is how far we are below the line. And in this case, it is negative one. | Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
We are told that a conservation group with a long-term goal of preserving species believes that all at-risk species will disappear when land inhabited by those species is developed. It has an opportunity to purchase land in an area about to be developed. The group has a choice of creating one large nature preserve with an area of 45 square kilometers and containing 70 at-risk species, or five small nature preserves, each with an area of three square kilometers and each containing 16 at-risk species unique to that preserve. Which choice would you recommend and why? And there's some interesting data here. Here it looks like some data they have gathered for different islands, and we have their areas, and then this is the number of species at risk in 1990, and then the species extinct by 2000. And so we can see for these various islands, we can see their areas and the proportion that got extinct, and it looks like they're plotted on this scatter plot. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
Which choice would you recommend and why? And there's some interesting data here. Here it looks like some data they have gathered for different islands, and we have their areas, and then this is the number of species at risk in 1990, and then the species extinct by 2000. And so we can see for these various islands, we can see their areas and the proportion that got extinct, and it looks like they're plotted on this scatter plot. Now be very careful when you look at this because look at the two axes. It is the vertical axes is the proportion extinct in 2000, so it's these numbers, but the horizontal axis isn't just a straight-up area. It's the natural log of the area. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
And so we can see for these various islands, we can see their areas and the proportion that got extinct, and it looks like they're plotted on this scatter plot. Now be very careful when you look at this because look at the two axes. It is the vertical axes is the proportion extinct in 2000, so it's these numbers, but the horizontal axis isn't just a straight-up area. It's the natural log of the area. And why did they do this? Well, notice, when you make the horizontal axis the natural log of the area, it looks like there's a linear relationship. But be clear, it's a linear relationship between the natural log of the area and proportion extinct in 2000. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
It's the natural log of the area. And why did they do this? Well, notice, when you make the horizontal axis the natural log of the area, it looks like there's a linear relationship. But be clear, it's a linear relationship between the natural log of the area and proportion extinct in 2000. But the reason why it's valuable to do this type of transformation is now we can apply our tools of linear regression to think about what would be the proportion extinct for the 45 square kilometers versus for the five small three-kilometer islands. So pause this video and see if you can figure it out on your own. And they give us the regression data for a line that fits this data. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
But be clear, it's a linear relationship between the natural log of the area and proportion extinct in 2000. But the reason why it's valuable to do this type of transformation is now we can apply our tools of linear regression to think about what would be the proportion extinct for the 45 square kilometers versus for the five small three-kilometer islands. So pause this video and see if you can figure it out on your own. And they give us the regression data for a line that fits this data. All right, now let's work through it together and to make some space because all of it is already plotted right over here and we have our regression data. So the regression line, we know it's slope and y-intercept. The y-intercept is right over here, 0.28996. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
And they give us the regression data for a line that fits this data. All right, now let's work through it together and to make some space because all of it is already plotted right over here and we have our regression data. So the regression line, we know it's slope and y-intercept. The y-intercept is right over here, 0.28996. So 0.2, this is, let's see, one, two, three, four, five, so 28996. It's almost two nine. So it's gonna be right over here would be the y-intercept. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
The y-intercept is right over here, 0.28996. So 0.2, this is, let's see, one, two, three, four, five, so 28996. It's almost two nine. So it's gonna be right over here would be the y-intercept. And its slope is negative 0.05 approximately. And I could eyeball it. It probably is going to look something like this. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
So it's gonna be right over here would be the y-intercept. And its slope is negative 0.05 approximately. And I could eyeball it. It probably is going to look something like this. That's the regression line. Or another way to think about it is the regression line tells us in general the proportion, proportion and I'll just say proportion shorthand for proportion extinct is going to be equal to our y-intercept, 0.28996 minus, minus 0.05323. And we have to be careful here. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
It probably is going to look something like this. That's the regression line. Or another way to think about it is the regression line tells us in general the proportion, proportion and I'll just say proportion shorthand for proportion extinct is going to be equal to our y-intercept, 0.28996 minus, minus 0.05323. And we have to be careful here. You might be tempted to say times the area, but no, the horizontal axis here is the natural log of the area, times the natural log of the area. And so we can use this equation for both scenarios to think about what is going to be the proportion that we would expect to get extinct in either situation. And then how many actual species will get extinct. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
And we have to be careful here. You might be tempted to say times the area, but no, the horizontal axis here is the natural log of the area, times the natural log of the area. And so we can use this equation for both scenarios to think about what is going to be the proportion that we would expect to get extinct in either situation. And then how many actual species will get extinct. And then the one that maybe has fewer species that get extinct is maybe the best one. Or the one that the more that we can preserve is maybe the best one. And so let's look at the two scenarios. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
And then how many actual species will get extinct. And then the one that maybe has fewer species that get extinct is maybe the best one. Or the one that the more that we can preserve is maybe the best one. And so let's look at the two scenarios. So the first scenario is the 45 square kilometer island. And this is just one, so times one. And so what is gonna be the proportion, proportion that we would expect to go extinct based on this regression? | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
And so let's look at the two scenarios. So the first scenario is the 45 square kilometer island. And this is just one, so times one. And so what is gonna be the proportion, proportion that we would expect to go extinct based on this regression? Well it's going to be 0.28996 minus 0.05323 times the natural log of 45. And if we wanna know the actual number that go extinct, so number extinct, extinct, would be equal to the proportion, would be equal to the proportion times, how many, let's see, the 45 square kilometers and it contains 70 at-risk species. So times our 70 species. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
And so what is gonna be the proportion, proportion that we would expect to go extinct based on this regression? Well it's going to be 0.28996 minus 0.05323 times the natural log of 45. And if we wanna know the actual number that go extinct, so number extinct, extinct, would be equal to the proportion, would be equal to the proportion times, how many, let's see, the 45 square kilometers and it contains 70 at-risk species. So times our 70 species. And so we can get our calculator out to figure that out. So this is the proportion we would expect to go extinct in the 45 square kilometer island based on our linear regression. So this would be equal to, so it looks like almost 9%. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
So times our 70 species. And so we can get our calculator out to figure that out. So this is the proportion we would expect to go extinct in the 45 square kilometer island based on our linear regression. So this would be equal to, so it looks like almost 9%. And if we wanna figure out the actual number we would expect to go extinct, we would just multiply that times the number of species on that island. So times 70. And we get, so approximately about 6.11. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
So this would be equal to, so it looks like almost 9%. And if we wanna figure out the actual number we would expect to go extinct, we would just multiply that times the number of species on that island. So times 70. And we get, so approximately about 6.11. So let me write that down. So this is going to be approximately 6.11. So we could say there would be approximately, if we, let's just say six extinct, this is all very approximate, extinct, and approximately 64 saved. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
And we get, so approximately about 6.11. So let me write that down. So this is going to be approximately 6.11. So we could say there would be approximately, if we, let's just say six extinct, this is all very approximate, extinct, and approximately 64 saved. Now let's think about the other scenario. Let's think about the scenario where we have five small nature preserves. So it's going to be three square kilometers times five islands. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
So we could say there would be approximately, if we, let's just say six extinct, this is all very approximate, extinct, and approximately 64 saved. Now let's think about the other scenario. Let's think about the scenario where we have five small nature preserves. So it's going to be three square kilometers times five islands. And we're gonna just do the same exercise. Our proportion that goes extinct is going to be 0.28996. That's just the y-intercept for our regression line. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
So it's going to be three square kilometers times five islands. And we're gonna just do the same exercise. Our proportion that goes extinct is going to be 0.28996. That's just the y-intercept for our regression line. Minus 0.05323, and I have a negative sign there because we have a negative slope. And this is not just times the area, it's times the natural log of the area. It's going to be three square kilometers. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
That's just the y-intercept for our regression line. Minus 0.05323, and I have a negative sign there because we have a negative slope. And this is not just times the area, it's times the natural log of the area. It's going to be three square kilometers. Three square kilometers. And then our number extinct, our number extinct is going to be equal to our proportion that we will calculate in the line above times, let's see, five small nature preserves, each with an area of three square kilometers and each containing 16 at-risk species. So five times 16, if each island has 16 and there's five islands, that's going to be five times 16 is 80. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
It's going to be three square kilometers. Three square kilometers. And then our number extinct, our number extinct is going to be equal to our proportion that we will calculate in the line above times, let's see, five small nature preserves, each with an area of three square kilometers and each containing 16 at-risk species. So five times 16, if each island has 16 and there's five islands, that's going to be five times 16 is 80. So times 80. So let's figure out what this is, get the calculator out again. And we are going to get, so this is going to be the proportion. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
So five times 16, if each island has 16 and there's five islands, that's going to be five times 16 is 80. So times 80. So let's figure out what this is, get the calculator out again. And we are going to get, so this is going to be the proportion. It's a much higher proportion. And then we'll multiply that times our number of species. So times 80 to figure out how many species will go extinct. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
And we are going to get, so this is going to be the proportion. It's a much higher proportion. And then we'll multiply that times our number of species. So times 80 to figure out how many species will go extinct. And we have here, it's approximately 18.52. So this is approximately 18.52. So another way to think about it is we're going to have approximately, well, if we round, let's just say 19 extinct, 19 extinct. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
So times 80 to figure out how many species will go extinct. And we have here, it's approximately 18.52. So this is approximately 18.52. So another way to think about it is we're going to have approximately, well, if we round, let's just say 19 extinct, 19 extinct. And then if we have 19 extinct, how many are we going to save? We're going to have 61 saved. 61 saved. | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
So another way to think about it is we're going to have approximately, well, if we round, let's just say 19 extinct, 19 extinct. And then if we have 19 extinct, how many are we going to save? We're going to have 61 saved. 61 saved. And even if you said 18 1⁄2 here and 61.5 here, on either measure, the 45 square, the big island is better. You're going to have fewer species that are extinct and more that are saved. So which choice would you recommend and why? | Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3 |
The school nurse plans to provide additional screening to students whose resting pulse rates are in the top 30% of the students who were tested. What is the minimum resting pulse rate at that school for students who will receive additional screening, round to the nearest whole number? If you feel like you know how to tackle this, I encourage you to pause this video and try to work it out. All right, now let's work this out together. They're telling us that the distribution of resting pulse rates are approximately normal. So we could use a normal distribution, and they tell us several things about this normal distribution. They tell us that the mean is 80 beats per minute, so that is the mean right over there, and they tell us that the standard deviation is nine beats per minute. | Finding z-score for a percentile AP Statistics Khan Academy.mp3 |
All right, now let's work this out together. They're telling us that the distribution of resting pulse rates are approximately normal. So we could use a normal distribution, and they tell us several things about this normal distribution. They tell us that the mean is 80 beats per minute, so that is the mean right over there, and they tell us that the standard deviation is nine beats per minute. So on this normal distribution, we have one standard deviation above the mean, two standard deviations above the mean, so this distance right over here is nine, so this would be 89. This one right over here would be 98, and you could also go standard deviations below the mean. This right over here would be 71. | Finding z-score for a percentile AP Statistics Khan Academy.mp3 |
They tell us that the mean is 80 beats per minute, so that is the mean right over there, and they tell us that the standard deviation is nine beats per minute. So on this normal distribution, we have one standard deviation above the mean, two standard deviations above the mean, so this distance right over here is nine, so this would be 89. This one right over here would be 98, and you could also go standard deviations below the mean. This right over here would be 71. This would be 62, but what we're concerned about is the top 30% because that is who is going to be tested. So there's gonna be some value here, some threshold. Let's say it is right over here, that if you are in that, if you are at that score, you have reached the minimum threshold to get additional screening. | Finding z-score for a percentile AP Statistics Khan Academy.mp3 |
This right over here would be 71. This would be 62, but what we're concerned about is the top 30% because that is who is going to be tested. So there's gonna be some value here, some threshold. Let's say it is right over here, that if you are in that, if you are at that score, you have reached the minimum threshold to get additional screening. You are in the top 30%. So that means that this area right over here is going to be 30%, or 0.3. So what we can do, we can use a z-table to say for what z-score is 70% of the distribution less than that, and then we can take that z-score and use the mean and the standard deviation to come up with an actual value. | Finding z-score for a percentile AP Statistics Khan Academy.mp3 |
Let's say it is right over here, that if you are in that, if you are at that score, you have reached the minimum threshold to get additional screening. You are in the top 30%. So that means that this area right over here is going to be 30%, or 0.3. So what we can do, we can use a z-table to say for what z-score is 70% of the distribution less than that, and then we can take that z-score and use the mean and the standard deviation to come up with an actual value. In previous examples, we started with a z-score and we're looking for the percentage. This time, we're looking for the percentage. We want it to be at least 70% and then come up with the corresponding z-score. | Finding z-score for a percentile AP Statistics Khan Academy.mp3 |
So what we can do, we can use a z-table to say for what z-score is 70% of the distribution less than that, and then we can take that z-score and use the mean and the standard deviation to come up with an actual value. In previous examples, we started with a z-score and we're looking for the percentage. This time, we're looking for the percentage. We want it to be at least 70% and then come up with the corresponding z-score. So let's see, immediately when we look at this, and we are to the right of the mean, and so we're gonna have a positive z-score, so we're starting at 50% here. We definitely wanna get, this is 67%, 68, 69. We're getting close, and on our table, this is the lowest z-score that gets us across that 70% threshold. | Finding z-score for a percentile AP Statistics Khan Academy.mp3 |
We want it to be at least 70% and then come up with the corresponding z-score. So let's see, immediately when we look at this, and we are to the right of the mean, and so we're gonna have a positive z-score, so we're starting at 50% here. We definitely wanna get, this is 67%, 68, 69. We're getting close, and on our table, this is the lowest z-score that gets us across that 70% threshold. It's at 0.7019. So it definitely crosses the threshold, and so that is a z-score of 0.53. 0.52 is too little. | Finding z-score for a percentile AP Statistics Khan Academy.mp3 |
We're getting close, and on our table, this is the lowest z-score that gets us across that 70% threshold. It's at 0.7019. So it definitely crosses the threshold, and so that is a z-score of 0.53. 0.52 is too little. So we need a z-score of 0.53. Let's write that down. 0.53, right over there, and we just now have to figure out what value gives us a z-score of 0.53. | Finding z-score for a percentile AP Statistics Khan Academy.mp3 |
0.52 is too little. So we need a z-score of 0.53. Let's write that down. 0.53, right over there, and we just now have to figure out what value gives us a z-score of 0.53. Well, this just means 0.53 standard deviations above the mean. So to get the value, we would take our mean, and we would add 0.53 standard deviation, so 0.53 times nine, and this will get us 0.53 times nine is equal to 4.77, plus 80 is equal to 84.77. 84.77, and they want us to round to the nearest whole number. | Finding z-score for a percentile AP Statistics Khan Academy.mp3 |
0.53, right over there, and we just now have to figure out what value gives us a z-score of 0.53. Well, this just means 0.53 standard deviations above the mean. So to get the value, we would take our mean, and we would add 0.53 standard deviation, so 0.53 times nine, and this will get us 0.53 times nine is equal to 4.77, plus 80 is equal to 84.77. 84.77, and they want us to round to the nearest whole number. So we will just round to 85 beats per minute. So that's the threshold. If you have that resting heartbeat, then the school nurse is going to give you some additional screening. | Finding z-score for a percentile AP Statistics Khan Academy.mp3 |
A hand is chosen, a hand is a collection of 9 cards, which can be sorted however the player chooses. Fair enough. How many 9 card hands are possible? So let's think about it. There are 36 unique cards, and I won't worry about there's 9 numbers in each suit, and there are 4 suits, 4 times 9 is 36. But let's just think of the cards as being 1 through 36, and we're going to pick 9 of them. So at first we'll say, well look, I have 9 slots in my hand, 1, 2, 3, 4, 5, 6, 7, 8, 9. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
So let's think about it. There are 36 unique cards, and I won't worry about there's 9 numbers in each suit, and there are 4 suits, 4 times 9 is 36. But let's just think of the cards as being 1 through 36, and we're going to pick 9 of them. So at first we'll say, well look, I have 9 slots in my hand, 1, 2, 3, 4, 5, 6, 7, 8, 9. I'm going to pick 9 cards for my hand. And so for the very first card, how many possible cards can I pick from? Well, there's 36 unique cards, so for that first slot, there's 36. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
So at first we'll say, well look, I have 9 slots in my hand, 1, 2, 3, 4, 5, 6, 7, 8, 9. I'm going to pick 9 cards for my hand. And so for the very first card, how many possible cards can I pick from? Well, there's 36 unique cards, so for that first slot, there's 36. But then that's now part of my hand. Now for the second slot, how many will there be left to pick from? Well, I've already picked 1, so there'll only be 35 to pick from, and then for the third slot, 34, and then it just keeps going, then 33 to pick from, 32, 31, 30, 29, and 28. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
Well, there's 36 unique cards, so for that first slot, there's 36. But then that's now part of my hand. Now for the second slot, how many will there be left to pick from? Well, I've already picked 1, so there'll only be 35 to pick from, and then for the third slot, 34, and then it just keeps going, then 33 to pick from, 32, 31, 30, 29, and 28. So you might want to say that there are 36 times 35 times 34 times 33 times 32 times 31 times 30 times 29 times 28 possible hands. Now, this would be true if order mattered. This would be true if maybe I have card 15 here. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
Well, I've already picked 1, so there'll only be 35 to pick from, and then for the third slot, 34, and then it just keeps going, then 33 to pick from, 32, 31, 30, 29, and 28. So you might want to say that there are 36 times 35 times 34 times 33 times 32 times 31 times 30 times 29 times 28 possible hands. Now, this would be true if order mattered. This would be true if maybe I have card 15 here. Maybe I have a 9 of spades here, and then I have a bunch of cards. And maybe I have, and that's one hand, and then I have another, so then I have cards 1, 2, 3, 4, 5, 6, 7, 8. I have 8 other cards. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
This would be true if maybe I have card 15 here. Maybe I have a 9 of spades here, and then I have a bunch of cards. And maybe I have, and that's one hand, and then I have another, so then I have cards 1, 2, 3, 4, 5, 6, 7, 8. I have 8 other cards. Or maybe another hand is I have the 8 cards, 1, 2, 3, 4, 5, 6, 7, 8, and then I have the 9 of spades. If we were thinking of these as two different hands, because we have the exact same cards, but they're in different order, then what I just calculated would make a lot of sense, because we did it based on order. But they're telling us that the cards can be sorted however the player chooses. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
I have 8 other cards. Or maybe another hand is I have the 8 cards, 1, 2, 3, 4, 5, 6, 7, 8, and then I have the 9 of spades. If we were thinking of these as two different hands, because we have the exact same cards, but they're in different order, then what I just calculated would make a lot of sense, because we did it based on order. But they're telling us that the cards can be sorted however the player chooses. So order doesn't matter. So we're over-counting. We're counting all of the different ways that the same number of cards can be arranged. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
But they're telling us that the cards can be sorted however the player chooses. So order doesn't matter. So we're over-counting. We're counting all of the different ways that the same number of cards can be arranged. So in order to, I guess, not over-count, we have to divide this by the way 9 cards can be rearranged. So how many ways can 9 cards be rearranged? If I have 9 cards, and I'm going to pick one of 9 to be in the first slot, well, that means I have 9 ways to put something in the first slot. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
We're counting all of the different ways that the same number of cards can be arranged. So in order to, I guess, not over-count, we have to divide this by the way 9 cards can be rearranged. So how many ways can 9 cards be rearranged? If I have 9 cards, and I'm going to pick one of 9 to be in the first slot, well, that means I have 9 ways to put something in the first slot. Then in the second slot, I have 8 ways of putting a card in the second slot, because I took 1 to put in the first, so I have 8 left, then 7, then 6, then 5, then 4, then 3, then 2, then 1. That last slot, there's only going to be one card left to put in it. So this number right here, where you take 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1, or 9, you start with 9, and then you multiply it by every number less than 9, every, I guess we could say, natural number less than 9, this is called 9 factorial. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
If I have 9 cards, and I'm going to pick one of 9 to be in the first slot, well, that means I have 9 ways to put something in the first slot. Then in the second slot, I have 8 ways of putting a card in the second slot, because I took 1 to put in the first, so I have 8 left, then 7, then 6, then 5, then 4, then 3, then 2, then 1. That last slot, there's only going to be one card left to put in it. So this number right here, where you take 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1, or 9, you start with 9, and then you multiply it by every number less than 9, every, I guess we could say, natural number less than 9, this is called 9 factorial. And you express it as an exclamation mark. So if we want to think about all of the different ways that we can have all of the different combinations for hands, this is the number of hands, if we cared about the order, but then we want to divide by the number of ways we can order things so that we don't overcount. And this will be an answer. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
So this number right here, where you take 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1, or 9, you start with 9, and then you multiply it by every number less than 9, every, I guess we could say, natural number less than 9, this is called 9 factorial. And you express it as an exclamation mark. So if we want to think about all of the different ways that we can have all of the different combinations for hands, this is the number of hands, if we cared about the order, but then we want to divide by the number of ways we can order things so that we don't overcount. And this will be an answer. And this will be the correct answer. Now, this is a super, super, duper large number. Let's figure out how large of a number this is. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
And this will be an answer. And this will be the correct answer. Now, this is a super, super, duper large number. Let's figure out how large of a number this is. We have 36 times 35 times 34 times 33 times 32 times 31 times 30 times 29 times 28 divided by 9. Well, I could do it this way. I could put a parentheses. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
Let's figure out how large of a number this is. We have 36 times 35 times 34 times 33 times 32 times 31 times 30 times 29 times 28 divided by 9. Well, I could do it this way. I could put a parentheses. Divided by parentheses 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1. Now, hopefully the calculator can handle this. And it gave us this number. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
I could put a parentheses. Divided by parentheses 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1. Now, hopefully the calculator can handle this. And it gave us this number. Was it 94,143,280? Let me put this on the side so I can read it. So this number right here gives us 94,143,280. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
And it gave us this number. Was it 94,143,280? Let me put this on the side so I can read it. So this number right here gives us 94,143,280. So that's the answer for this problem. That there are 94,143,280 possible 9-card hands in this situation. Now, we kind of just worked through it. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
So this number right here gives us 94,143,280. So that's the answer for this problem. That there are 94,143,280 possible 9-card hands in this situation. Now, we kind of just worked through it. We reasoned our way through it. There is a formula for this that does essentially the exact same thing. And the way that people denote this formula is they say, look, we have 36 things. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
Now, we kind of just worked through it. We reasoned our way through it. There is a formula for this that does essentially the exact same thing. And the way that people denote this formula is they say, look, we have 36 things. And we are going to choose 9 of them. We don't care about order. So sometimes it'll be written as n choose k. Let me write it this way. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
And the way that people denote this formula is they say, look, we have 36 things. And we are going to choose 9 of them. We don't care about order. So sometimes it'll be written as n choose k. Let me write it this way. So what did we do here? We have 36 things. We chose 9. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
So sometimes it'll be written as n choose k. Let me write it this way. So what did we do here? We have 36 things. We chose 9. So this numerator over here, this was 36 factorial. But 36 factorial would go all the way down to 27, 26, 25. It would just keep going. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
We chose 9. So this numerator over here, this was 36 factorial. But 36 factorial would go all the way down to 27, 26, 25. It would just keep going. But we stopped only 9 away from 36. So this is 36 factorial. So this part right here, that part right there, is not just 36 factorial. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
It would just keep going. But we stopped only 9 away from 36. So this is 36 factorial. So this part right here, that part right there, is not just 36 factorial. It's 36 factorial divided by 36 minus 9 factorial. What is 36 minus 9? It's 27. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
So this part right here, that part right there, is not just 36 factorial. It's 36 factorial divided by 36 minus 9 factorial. What is 36 minus 9? It's 27. So 27 factorial. So let's think about this. 36 factorial, it'd be 36 times 35. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
It's 27. So 27 factorial. So let's think about this. 36 factorial, it'd be 36 times 35. You keep going. All the way times 28 times 27, all the way down to 1. That is 36 factorial. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
36 factorial, it'd be 36 times 35. You keep going. All the way times 28 times 27, all the way down to 1. That is 36 factorial. Now what is 36 minus 9 factorial? That's 27 factorial. So if you divide by 27 factorial, 27 factorial is 27 times 26, all the way down to 1. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
That is 36 factorial. Now what is 36 minus 9 factorial? That's 27 factorial. So if you divide by 27 factorial, 27 factorial is 27 times 26, all the way down to 1. Well, this and this are the exact same thing. This is 27 times 26. So that and that would cancel out. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
So if you divide by 27 factorial, 27 factorial is 27 times 26, all the way down to 1. Well, this and this are the exact same thing. This is 27 times 26. So that and that would cancel out. So if you do 36 divided by 36 minus 9 factorial, you just get the first, I guess the largest 9 terms of 36 factorial, which is exactly what we have over there. So that is that. And then we divided it by 9 factorial. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
So that and that would cancel out. So if you do 36 divided by 36 minus 9 factorial, you just get the first, I guess the largest 9 terms of 36 factorial, which is exactly what we have over there. So that is that. And then we divided it by 9 factorial. And we divided it by 9 factorial. And this right here is called 36 choose 9. And sometimes you'll see this formula written like this. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
And then we divided it by 9 factorial. And we divided it by 9 factorial. And this right here is called 36 choose 9. And sometimes you'll see this formula written like this. n choose k, and they'll write the formula as equal to n factorial over n minus k factorial. And also in the denominator, k factorial. And this is a general formula that if you have n things and you want to find out all of the possible ways you can pick k things from those n things and you don't care about the order. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
And sometimes you'll see this formula written like this. n choose k, and they'll write the formula as equal to n factorial over n minus k factorial. And also in the denominator, k factorial. And this is a general formula that if you have n things and you want to find out all of the possible ways you can pick k things from those n things and you don't care about the order. All you care is about which k things you picked. You don't care about the order in which you picked those k things. So that's what we did here. | Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3 |
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. 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. | Example Different ways to pick officers Precalculus Khan Academy.mp3 |
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. 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. | Example Different ways to pick officers Precalculus Khan Academy.mp3 |
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. 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. | Example Different ways to pick officers Precalculus Khan Academy.mp3 |
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. So to answer the question, the probability of Marsha being president, Savita being vice president, Robert being secretary, is 1 over the total number of possibilities, which is 1 over 504. That's the probability. | Example Different ways to pick officers Precalculus Khan Academy.mp3 |
Each problem has only one correct answer. What is the probability of randomly guessing the correct answer on both problems? Now, the probability of guessing the correct answer on each problem, these are independent events. So let's write this down. The probability of correct on problem number one is independent. Or let me write it this way. Probability of correct on number one and probability of correct on problem two are independent. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
So let's write this down. The probability of correct on problem number one is independent. Or let me write it this way. Probability of correct on number one and probability of correct on problem two are independent. Independent. Which means that the outcome of one of the events of guessing on the first problem isn't going to affect the probability of guessing correctly on the second problem, independent events. So the probability of guessing on both of them, so that means that the probability of being correct on one and number two is going to be equal to the product of these probabilities. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
Probability of correct on number one and probability of correct on problem two are independent. Independent. Which means that the outcome of one of the events of guessing on the first problem isn't going to affect the probability of guessing correctly on the second problem, independent events. So the probability of guessing on both of them, so that means that the probability of being correct on one and number two is going to be equal to the product of these probabilities. And we're going to see why that is visually in a second. But it's going to be the probability of correct on number one times the probability of being correct on number two. Now, what are each of these probabilities? | Test taking probability and independent events Precalculus Khan Academy.mp3 |
So the probability of guessing on both of them, so that means that the probability of being correct on one and number two is going to be equal to the product of these probabilities. And we're going to see why that is visually in a second. But it's going to be the probability of correct on number one times the probability of being correct on number two. Now, what are each of these probabilities? On number one, there are four choices. There are four possible outcomes. And only one of them is going to be correct. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
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