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And I'm just playing around with the formula, because I really want you to see all the ways that it, you know, maybe you'll get a little intuition. And I encourage you to email me if you see some insight on why this exists and all of that. But once again, I think it is cool that all of a sudden we have this other formula that has pi and e in it, and this is really just, you know, this is what the central, you know, so many phenomenon are described by this. And once again, pi and e show up together, right? Just like e to the i pi is equal to negative 1 tells you something about our universe. But anyway, I could rewrite this as e to the x minus mu over sigma squared, and all of that to the minus 1 half. Something to the minus 1 half power, that's just 1 over the square root, which is already going on here. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And once again, pi and e show up together, right? Just like e to the i pi is equal to negative 1 tells you something about our universe. But anyway, I could rewrite this as e to the x minus mu over sigma squared, and all of that to the minus 1 half. Something to the minus 1 half power, that's just 1 over the square root, which is already going on here. So we could just rewrite this over here as 1 over the square root of 2 pi times the variance times e to essentially our z-score squared, right? If we say z is this thing in here, z is how many standard deviations we are from the mean, z-score squared. And all of a sudden this kind of becomes a very clean, you know, we just say 2 pi times our variance times e to the number of standard deviations we are away from the mean. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Something to the minus 1 half power, that's just 1 over the square root, which is already going on here. So we could just rewrite this over here as 1 over the square root of 2 pi times the variance times e to essentially our z-score squared, right? If we say z is this thing in here, z is how many standard deviations we are from the mean, z-score squared. And all of a sudden this kind of becomes a very clean, you know, we just say 2 pi times our variance times e to the number of standard deviations we are away from the mean. You square that. You take the square root of that thing and invert it, and that's also the normal distribution. So anyway, I wanted to do that just because I thought it was neat and it's interesting to play around with it, and that way if you see it in any of these other forms in the rest of your life, you won't say, what's that? | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And all of a sudden this kind of becomes a very clean, you know, we just say 2 pi times our variance times e to the number of standard deviations we are away from the mean. You square that. You take the square root of that thing and invert it, and that's also the normal distribution. So anyway, I wanted to do that just because I thought it was neat and it's interesting to play around with it, and that way if you see it in any of these other forms in the rest of your life, you won't say, what's that? I thought the normal distribution was this or was this, and now you know. With that said, let's play around a little bit with this normal distribution. So in this spreadsheet, I've plotted normal distribution. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So anyway, I wanted to do that just because I thought it was neat and it's interesting to play around with it, and that way if you see it in any of these other forms in the rest of your life, you won't say, what's that? I thought the normal distribution was this or was this, and now you know. With that said, let's play around a little bit with this normal distribution. So in this spreadsheet, I've plotted normal distribution. You can change the assumptions that are in this kind of a green-blue color. So right now it's plotting it with a mean of 0 and a standard deviation of 4. And I just write the variance here just for your information. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So in this spreadsheet, I've plotted normal distribution. You can change the assumptions that are in this kind of a green-blue color. So right now it's plotting it with a mean of 0 and a standard deviation of 4. And I just write the variance here just for your information. The variance is just the standard deviation squared. And so what happens when you change the mean? So if the mean goes from 0 to let's say it goes to 5. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And I just write the variance here just for your information. The variance is just the standard deviation squared. And so what happens when you change the mean? So if the mean goes from 0 to let's say it goes to 5. Notice this graph just shifted to the right by 5, right? It was centered here. Now it's centered over here. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So if the mean goes from 0 to let's say it goes to 5. Notice this graph just shifted to the right by 5, right? It was centered here. Now it's centered over here. If we make it minus 5, what happens? The whole bell curve just shifts 5 to the left from the center. Now what happens when you change the standard deviation? | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Now it's centered over here. If we make it minus 5, what happens? The whole bell curve just shifts 5 to the left from the center. Now what happens when you change the standard deviation? The standard deviation is a measure of the average squared distance from the mean. The standard deviation is the square root of that. So it's kind of, not exactly, but kind of the average distance from the mean. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Now what happens when you change the standard deviation? The standard deviation is a measure of the average squared distance from the mean. The standard deviation is the square root of that. So it's kind of, not exactly, but kind of the average distance from the mean. So the smaller the standard deviation, the closer a lot of the points are going to be to the mean. So we should get kind of a narrower graph. And let's see that happens. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So it's kind of, not exactly, but kind of the average distance from the mean. So the smaller the standard deviation, the closer a lot of the points are going to be to the mean. So we should get kind of a narrower graph. And let's see that happens. So when the standard deviation is 2, we see that. The graph, you're more likely to be really close to the mean than further away. And if you make the standard deviation, I don't know, if you make it 10, all of a sudden you get a really flat graph. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And let's see that happens. So when the standard deviation is 2, we see that. The graph, you're more likely to be really close to the mean than further away. And if you make the standard deviation, I don't know, if you make it 10, all of a sudden you get a really flat graph. And this thing keeps going on forever. And that's a key difference. The binomial distribution is always finite. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And if you make the standard deviation, I don't know, if you make it 10, all of a sudden you get a really flat graph. And this thing keeps going on forever. And that's a key difference. The binomial distribution is always finite. You can only have a finite number of values. While the normal distribution is defined over the entire real number line. So the probability, if you have a mean of minus 5 and a standard deviation of 10, the probability of getting 1,000 here is very, very low. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
The binomial distribution is always finite. You can only have a finite number of values. While the normal distribution is defined over the entire real number line. So the probability, if you have a mean of minus 5 and a standard deviation of 10, the probability of getting 1,000 here is very, very low. But there is some probability that all of the atoms in my body just arrange perfectly that I fall through the seat I'm sitting on. It's very unlikely. And it probably won't happen in the life of the universe. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So the probability, if you have a mean of minus 5 and a standard deviation of 10, the probability of getting 1,000 here is very, very low. But there is some probability that all of the atoms in my body just arrange perfectly that I fall through the seat I'm sitting on. It's very unlikely. And it probably won't happen in the life of the universe. But it can happen. And that could be described by a normal distribution. Because it says anything can happen, although it could be very, very, very unprobable. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And it probably won't happen in the life of the universe. But it can happen. And that could be described by a normal distribution. Because it says anything can happen, although it could be very, very, very unprobable. So the thing I talked about at the beginning of the video is when you figure out a normal distribution, you can't just look at this point on the graph. Let me get the pen tool back. You have to figure out the area under the curve between two points. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Because it says anything can happen, although it could be very, very, very unprobable. So the thing I talked about at the beginning of the video is when you figure out a normal distribution, you can't just look at this point on the graph. Let me get the pen tool back. You have to figure out the area under the curve between two points. So if I wanted to say, let's say this was our distribution. I said, what is the probability that I get 0? I don't know what phenomena this is describing, but that 0 happened. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
You have to figure out the area under the curve between two points. So if I wanted to say, let's say this was our distribution. I said, what is the probability that I get 0? I don't know what phenomena this is describing, but that 0 happened. If I say exactly 0, the probability is 0. Because I shouldn't use 0 too much. Because the area under the curve, just under 0, there's no area. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
I don't know what phenomena this is describing, but that 0 happened. If I say exactly 0, the probability is 0. Because I shouldn't use 0 too much. Because the area under the curve, just under 0, there's no area. It's just a line. You have to say between a range. So you have to say the probability between, let's say minus, and actually I can type it in here. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Because the area under the curve, just under 0, there's no area. It's just a line. You have to say between a range. So you have to say the probability between, let's say minus, and actually I can type it in here. I can say the probability between, let's say, minus 0.005 and plus 0.05 is, well, it rounded. So it says they're close to 0. Let me do it. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So you have to say the probability between, let's say minus, and actually I can type it in here. I can say the probability between, let's say, minus 0.005 and plus 0.05 is, well, it rounded. So it says they're close to 0. Let me do it. Between minus 1 and between 1. It calculated it at 7%. And I'll show you how I calculated this in a second. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Let me do it. Between minus 1 and between 1. It calculated it at 7%. And I'll show you how I calculated this in a second. So let me get the screen drawn to a log. So what did I just do? This, between minus 1 and 1, and I'll show you behind the scenes what Excel is doing. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And I'll show you how I calculated this in a second. So let me get the screen drawn to a log. So what did I just do? This, between minus 1 and 1, and I'll show you behind the scenes what Excel is doing. We're going from minus 1, which is roughly right here, to 1, and we're calculating the area under the curve. We're calculating this area. Or for those of you who know calculus, we're calculating the integral from minus 1 to 1 of this function, where the standard deviation is right here, is 10, and the mean is minus 5. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
This, between minus 1 and 1, and I'll show you behind the scenes what Excel is doing. We're going from minus 1, which is roughly right here, to 1, and we're calculating the area under the curve. We're calculating this area. Or for those of you who know calculus, we're calculating the integral from minus 1 to 1 of this function, where the standard deviation is right here, is 10, and the mean is minus 5. Actually, let me put that in. So we're calculating, for this example, the way it's drawn right here, the normal distribution function, let's see, our standard deviation is 10 times the square root of 2 pi times e to the minus 1 half times x minus our mean. Our mean is negative right now. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Or for those of you who know calculus, we're calculating the integral from minus 1 to 1 of this function, where the standard deviation is right here, is 10, and the mean is minus 5. Actually, let me put that in. So we're calculating, for this example, the way it's drawn right here, the normal distribution function, let's see, our standard deviation is 10 times the square root of 2 pi times e to the minus 1 half times x minus our mean. Our mean is negative right now. Our mean is minus 5, so it's x plus 5 over the standard deviation squared, which is the variance. So that's 100 squared dx. This is what this number is right here, this 7%, or actually 0.07 is the area right under there. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Our mean is negative right now. Our mean is minus 5, so it's x plus 5 over the standard deviation squared, which is the variance. So that's 100 squared dx. This is what this number is right here, this 7%, or actually 0.07 is the area right under there. Now, unfortunately for us in the world, this isn't an easy integral to evaluate analytically, even for those of us who know our calculus. So this tends to be done numerically, and kind of an easy way to do this, well, not an easy way, but a function has been defined called the cumulative distribution function that is a useful tool for figuring out this area. So what the cumulative distribution function is essentially, let me call it, cumulative distribution function, it's a function of x. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
This is what this number is right here, this 7%, or actually 0.07 is the area right under there. Now, unfortunately for us in the world, this isn't an easy integral to evaluate analytically, even for those of us who know our calculus. So this tends to be done numerically, and kind of an easy way to do this, well, not an easy way, but a function has been defined called the cumulative distribution function that is a useful tool for figuring out this area. So what the cumulative distribution function is essentially, let me call it, cumulative distribution function, it's a function of x. It gives us the area under the curve, under this curve, so let's say that this is x right here, that's our x. It tells you the area under the curve up to x. Or so another way to think about it, it tells you what is the probability that you land at some value less than your x value, so it's the area from minus infinity to x of our probability density function, dx. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So what the cumulative distribution function is essentially, let me call it, cumulative distribution function, it's a function of x. It gives us the area under the curve, under this curve, so let's say that this is x right here, that's our x. It tells you the area under the curve up to x. Or so another way to think about it, it tells you what is the probability that you land at some value less than your x value, so it's the area from minus infinity to x of our probability density function, dx. And there's actually an Excel, when you actually use the Excel normal distribution function, you say norm distribution, you have to give it your x value, you give it the mean, you give it the standard deviation, and then you say whether you want the cumulative distribution, in which case you say true, or you want just this normal distribution, which you say false. So if you wanted to graph this right here, you would say false, in caps. If you wanted to graph the cumulative distribution function, which I do down here, let me move this down a little bit, let me get out of the pen tool, so the cumulative distribution function is right over here, then you say true when you make that Excel call. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Or so another way to think about it, it tells you what is the probability that you land at some value less than your x value, so it's the area from minus infinity to x of our probability density function, dx. And there's actually an Excel, when you actually use the Excel normal distribution function, you say norm distribution, you have to give it your x value, you give it the mean, you give it the standard deviation, and then you say whether you want the cumulative distribution, in which case you say true, or you want just this normal distribution, which you say false. So if you wanted to graph this right here, you would say false, in caps. If you wanted to graph the cumulative distribution function, which I do down here, let me move this down a little bit, let me get out of the pen tool, so the cumulative distribution function is right over here, then you say true when you make that Excel call. So this is a cumulative distribution function for the same, for this, this is a normal distribution, here's a cumulative distribution. And just so you get the intuition, if you want to know what is the probability that I get a value less than 20, so I can get any value less than 20 given this distribution, the cumulative distribution right here, let me make it so you can see the, if you go to 20, you just go right to that point there and you say wow, the probability of getting 20 or less, it's pretty high, it's approaching 100%, that makes sense because most of the area under this curve is less than 20. Or if you said what's the probability of getting less than minus 5, well minus 5 was the mean, so half of your result should be above that and half should be below. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
If you wanted to graph the cumulative distribution function, which I do down here, let me move this down a little bit, let me get out of the pen tool, so the cumulative distribution function is right over here, then you say true when you make that Excel call. So this is a cumulative distribution function for the same, for this, this is a normal distribution, here's a cumulative distribution. And just so you get the intuition, if you want to know what is the probability that I get a value less than 20, so I can get any value less than 20 given this distribution, the cumulative distribution right here, let me make it so you can see the, if you go to 20, you just go right to that point there and you say wow, the probability of getting 20 or less, it's pretty high, it's approaching 100%, that makes sense because most of the area under this curve is less than 20. Or if you said what's the probability of getting less than minus 5, well minus 5 was the mean, so half of your result should be above that and half should be below. And if you go to this point right here, you can see that this right here is 50%. So the probability of getting less than minus 5 is exactly 50%. So what you do is, if I wanted to know the probability of getting between negative 1 and 1, what I do is, let me get back to my pen tool, what I do is I figure out what is the probability of getting minus 1 or lower, right? | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Or if you said what's the probability of getting less than minus 5, well minus 5 was the mean, so half of your result should be above that and half should be below. And if you go to this point right here, you can see that this right here is 50%. So the probability of getting less than minus 5 is exactly 50%. So what you do is, if I wanted to know the probability of getting between negative 1 and 1, what I do is, let me get back to my pen tool, what I do is I figure out what is the probability of getting minus 1 or lower, right? So I figure out this whole area and then I figure out the probability of getting 1 or lower, which is this whole area, well let me do it in a different color, 1 or lower is everything there. And I subtract the yellow area from the magenta area and I'll just get what's ever left over here, right? So what I do is I take, and that's exactly what I did in the spreadsheet. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So what you do is, if I wanted to know the probability of getting between negative 1 and 1, what I do is, let me get back to my pen tool, what I do is I figure out what is the probability of getting minus 1 or lower, right? So I figure out this whole area and then I figure out the probability of getting 1 or lower, which is this whole area, well let me do it in a different color, 1 or lower is everything there. And I subtract the yellow area from the magenta area and I'll just get what's ever left over here, right? So what I do is I take, and that's exactly what I did in the spreadsheet. Let me scroll down. This might be taxing my computer by taking the screen capture with it. So what I did is I evaluated the cumulative distribution function at 1, which would be right there. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So what I do is I take, and that's exactly what I did in the spreadsheet. Let me scroll down. This might be taxing my computer by taking the screen capture with it. So what I did is I evaluated the cumulative distribution function at 1, which would be right there. And I evaluate the cumulative distribution function at minus 1, which is right there. And the difference between these two, I subtract this number from this number and that tells me essentially the probability that I'm between those two numbers, or another way to think about it, the area right here. And I really encourage you to play with this and explore the Excel formulas and everything. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So what I did is I evaluated the cumulative distribution function at 1, which would be right there. And I evaluate the cumulative distribution function at minus 1, which is right there. And the difference between these two, I subtract this number from this number and that tells me essentially the probability that I'm between those two numbers, or another way to think about it, the area right here. And I really encourage you to play with this and explore the Excel formulas and everything. This area right here, between minus 1 and 1. Now one thing that shows up a lot is what's the probability that you land within the standard deviation of, and just so you know this graph, the central line right here, this is the mean, and then these two lines I drew right here, these are one standard deviation below and one standard deviation above the mean. And some people think, what's the probability that I land within one standard deviation of the mean? | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And I really encourage you to play with this and explore the Excel formulas and everything. This area right here, between minus 1 and 1. Now one thing that shows up a lot is what's the probability that you land within the standard deviation of, and just so you know this graph, the central line right here, this is the mean, and then these two lines I drew right here, these are one standard deviation below and one standard deviation above the mean. And some people think, what's the probability that I land within one standard deviation of the mean? Well, that's easy to do. What I can do is I'll just click on this and I could call this, what's the probability that I land between, let's see, one standard deviation, the mean is minus 5, one standard deviation below the mean is minus 15. And one standard deviation above the mean is 10 plus minus 5 is 5. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And some people think, what's the probability that I land within one standard deviation of the mean? Well, that's easy to do. What I can do is I'll just click on this and I could call this, what's the probability that I land between, let's see, one standard deviation, the mean is minus 5, one standard deviation below the mean is minus 15. And one standard deviation above the mean is 10 plus minus 5 is 5. So that's between 5 and 15. So 68.3%, and that's actually always the case, that you have a 68.3% probability of landing within one standard deviation of the mean, assuming you have a normal distribution. So once again, that number comes from, that represents the area under the curve here. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And one standard deviation above the mean is 10 plus minus 5 is 5. So that's between 5 and 15. So 68.3%, and that's actually always the case, that you have a 68.3% probability of landing within one standard deviation of the mean, assuming you have a normal distribution. So once again, that number comes from, that represents the area under the curve here. This area under the curve. And the way you get it is with the cumulative distribution function. Let me go down here. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So once again, that number comes from, that represents the area under the curve here. This area under the curve. And the way you get it is with the cumulative distribution function. Let me go down here. Every time I move this, I have to get rid of the pen tool. So you go from, you evaluate it at plus 5, which is right here, this was one standard deviation above the mean, which, that's a number right around there. Looks like it's like, I don't know, 80 something percent, maybe 90% roughly. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Let me go down here. Every time I move this, I have to get rid of the pen tool. So you go from, you evaluate it at plus 5, which is right here, this was one standard deviation above the mean, which, that's a number right around there. Looks like it's like, I don't know, 80 something percent, maybe 90% roughly. And then you evaluate it at one standard deviation below the mean, which is minus 15. And this one looks like, I don't know, roughly 15% or so. 15%, 16%, maybe 17%, let's say 18%. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Looks like it's like, I don't know, 80 something percent, maybe 90% roughly. And then you evaluate it at one standard deviation below the mean, which is minus 15. And this one looks like, I don't know, roughly 15% or so. 15%, 16%, maybe 17%, let's say 18%. But the big picture is when you subtract this value from this value, you get the probability that you land between those two. And that's because this value tells the probability that you're less than. So when you go to the cumulative distribution function, you get that right there. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
15%, 16%, maybe 17%, let's say 18%. But the big picture is when you subtract this value from this value, you get the probability that you land between those two. And that's because this value tells the probability that you're less than. So when you go to the cumulative distribution function, you get that right there. That tells the probability that you are, let me get, I keep scrolling back and forth. Let me, that tells you that you're the, so when you go to 5, and you just go right over here, this essentially tells you this area under the curve. The probability that you're less than or equal to 5, everything up there. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So when you go to the cumulative distribution function, you get that right there. That tells the probability that you are, let me get, I keep scrolling back and forth. Let me, that tells you that you're the, so when you go to 5, and you just go right over here, this essentially tells you this area under the curve. The probability that you're less than or equal to 5, everything up there. And then when you evaluate it at minus 15 down here, it tells you the probability that you're down back here. So when you subtract this from the larger thing, you're just left with what's under the curve right there. And just to understand this spreadsheet a little bit better, just because I really want you to play with it and move the, see what happens when, if I make this distribution, the mean was minus 5, now let me make it 5. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
The probability that you're less than or equal to 5, everything up there. And then when you evaluate it at minus 15 down here, it tells you the probability that you're down back here. So when you subtract this from the larger thing, you're just left with what's under the curve right there. And just to understand this spreadsheet a little bit better, just because I really want you to play with it and move the, see what happens when, if I make this distribution, the mean was minus 5, now let me make it 5. It just shifted to the right. It just moved over to the right by 5, right? Whoops. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And just to understand this spreadsheet a little bit better, just because I really want you to play with it and move the, see what happens when, if I make this distribution, the mean was minus 5, now let me make it 5. It just shifted to the right. It just moved over to the right by 5, right? Whoops. I'll use the pen tool. It just moved over to the right by 5. If I were to try to make the standard deviation smaller, we'll see that the whole thing just gets a little bit tighter. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Whoops. I'll use the pen tool. It just moved over to the right by 5. If I were to try to make the standard deviation smaller, we'll see that the whole thing just gets a little bit tighter. Let's make it 6. And all of a sudden, this looks a little bit tighter curve, we make it 2, it becomes even tighter. And just so you know how I calculated everything, and I really want you to play with this and play with the formula and get an intuitive feeling for this, the cumulative distribution function, and think a lot about how it relates to the binomial distribution, and I covered that in the last video. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
If I were to try to make the standard deviation smaller, we'll see that the whole thing just gets a little bit tighter. Let's make it 6. And all of a sudden, this looks a little bit tighter curve, we make it 2, it becomes even tighter. And just so you know how I calculated everything, and I really want you to play with this and play with the formula and get an intuitive feeling for this, the cumulative distribution function, and think a lot about how it relates to the binomial distribution, and I covered that in the last video. To plot this, I just took each of these points. I went to plot the points between minus 20 and 20, and I just incremented by 1. I just decided to increment by 1. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And just so you know how I calculated everything, and I really want you to play with this and play with the formula and get an intuitive feeling for this, the cumulative distribution function, and think a lot about how it relates to the binomial distribution, and I covered that in the last video. To plot this, I just took each of these points. I went to plot the points between minus 20 and 20, and I just incremented by 1. I just decided to increment by 1. So this isn't a continuous curve. It's actually just plotting a point at each point and connecting it with a line. Then I did the distance between each of those points and the mean. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
I just decided to increment by 1. So this isn't a continuous curve. It's actually just plotting a point at each point and connecting it with a line. Then I did the distance between each of those points and the mean. So I just took, let's say, 0 minus 5. This is this distance. So this just tells you the point minus 20 is 25 less than the mean, right? | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Then I did the distance between each of those points and the mean. So I just took, let's say, 0 minus 5. This is this distance. So this just tells you the point minus 20 is 25 less than the mean, right? That's all I did there. Then I divided that by the standard deviation, and this is the z-score, the standard z-score. So this tells me how many standard deviations is minus 20 away from the mean. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So this just tells you the point minus 20 is 25 less than the mean, right? That's all I did there. Then I divided that by the standard deviation, and this is the z-score, the standard z-score. So this tells me how many standard deviations is minus 20 away from the mean. It's 12 and 1 half standard deviations below the mean. And then I used that, and I just plugged it into essentially this formula to figure out the height of the function. So let's say at minus 20, the height is very low. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So this tells me how many standard deviations is minus 20 away from the mean. It's 12 and 1 half standard deviations below the mean. And then I used that, and I just plugged it into essentially this formula to figure out the height of the function. So let's say at minus 20, the height is very low. At minus 5, well, let's say at minus 2, the height's a little bit better. The height's going to be someplace, it's going to be like right there. And so that gives me that value. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So let's say at minus 20, the height is very low. At minus 5, well, let's say at minus 2, the height's a little bit better. The height's going to be someplace, it's going to be like right there. And so that gives me that value. But then to actually figure out the probability of that, what I do is I calculate the cumulative distribution function between, well, this is the value, the probability that you're less than that. So the area under the curve below that, which is very, very small. It's not 0. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And so that gives me that value. But then to actually figure out the probability of that, what I do is I calculate the cumulative distribution function between, well, this is the value, the probability that you're less than that. So the area under the curve below that, which is very, very small. It's not 0. I know it looks like 0 here, but that's only because I round it. It's going to be 0, 0, 0, 1. It's going to be a really, really small number. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
It's not 0. I know it looks like 0 here, but that's only because I round it. It's going to be 0, 0, 0, 1. It's going to be a really, really small number. There's some probability that we even get like minus 1,000. And another intuitive thing that you really should have a sense for is the integral over this, or the entire area of the curve, has to be 1. Because that takes into account all possible circumstances. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
It's going to be a really, really small number. There's some probability that we even get like minus 1,000. And another intuitive thing that you really should have a sense for is the integral over this, or the entire area of the curve, has to be 1. Because that takes into account all possible circumstances. And that should happen if we put a suitably small number here and a suitably large number here. There you go. We get 100%. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Because that takes into account all possible circumstances. And that should happen if we put a suitably small number here and a suitably large number here. There you go. We get 100%. Although this isn't 100%. We would have to go from minus infinity to plus infinity to really get 100%. It's just rounding to 100%. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
We get 100%. Although this isn't 100%. We would have to go from minus infinity to plus infinity to really get 100%. It's just rounding to 100%. It's probably 99.999999% or something like that. And so to actually calculate this, what I do is I take the cumulative distribution function of this point, and I subtract from that the cumulative distribution function of that point. And that's where I got this 100% from. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
It's just rounding to 100%. It's probably 99.999999% or something like that. And so to actually calculate this, what I do is I take the cumulative distribution function of this point, and I subtract from that the cumulative distribution function of that point. And that's where I got this 100% from. Anyway, hopefully that'll give you a good feel for the normal distribution. And I really encourage you to play with the spreadsheet and to even make a spreadsheet like this yourself. And in future exercises, we'll actually use this type of spreadsheet as an input into other models. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And that's where I got this 100% from. Anyway, hopefully that'll give you a good feel for the normal distribution. And I really encourage you to play with the spreadsheet and to even make a spreadsheet like this yourself. And in future exercises, we'll actually use this type of spreadsheet as an input into other models. So if we're doing a financial model, and if we say our revenue has a normal distribution around some expected value, what is the distribution of our net income? Or we could think of 100 other different types of examples. Anyway, see you in the next video. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Assume that the conditions for inference were met. What is the approximate p-value for Katerina's test? So like always, pause this video and see if you can figure it out. Well, I just always like to remind ourselves what's going on here. So there's some population here. She has a null hypothesis that the mean is equal to zero, but the alternative is that it's not equal to zero. She wants to test her null hypothesis, so she takes a sample size, or she takes a sample of size six. | Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 |
Well, I just always like to remind ourselves what's going on here. So there's some population here. She has a null hypothesis that the mean is equal to zero, but the alternative is that it's not equal to zero. She wants to test her null hypothesis, so she takes a sample size, or she takes a sample of size six. From that, since we care about, the population parameter we care about is the population mean, she would calculate the sample mean in order to estimate that, and the sample standard deviation. And then from that, we can calculate this T-value. The T-value is going to be equal to the difference between her sample mean and the assumed, the assumed population mean from the null hypothesis, that's what this little sub zero means, it means it's the assumed mean from the null hypothesis, divided by our estimate of the standard deviation of the sampling distribution. | Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 |
She wants to test her null hypothesis, so she takes a sample size, or she takes a sample of size six. From that, since we care about, the population parameter we care about is the population mean, she would calculate the sample mean in order to estimate that, and the sample standard deviation. And then from that, we can calculate this T-value. The T-value is going to be equal to the difference between her sample mean and the assumed, the assumed population mean from the null hypothesis, that's what this little sub zero means, it means it's the assumed mean from the null hypothesis, divided by our estimate of the standard deviation of the sampling distribution. I say estimate because unlike when we were dealing with proportions, with proportions we can actually calculate the assumed based on the null hypothesis, sampling distribution standard deviation, but here we have to estimate it. And so it's going to be our sample standard deviation divided by the square root of N. Now in this example, they calculated all of this for us. They said, hey, this is going to be equal to 2.75. | Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 |
The T-value is going to be equal to the difference between her sample mean and the assumed, the assumed population mean from the null hypothesis, that's what this little sub zero means, it means it's the assumed mean from the null hypothesis, divided by our estimate of the standard deviation of the sampling distribution. I say estimate because unlike when we were dealing with proportions, with proportions we can actually calculate the assumed based on the null hypothesis, sampling distribution standard deviation, but here we have to estimate it. And so it's going to be our sample standard deviation divided by the square root of N. Now in this example, they calculated all of this for us. They said, hey, this is going to be equal to 2.75. And so we can just use that to figure out our P-value. But let's just think about what that is asking us to do. So the null hypothesis is that the mean is zero. | Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 |
They said, hey, this is going to be equal to 2.75. And so we can just use that to figure out our P-value. But let's just think about what that is asking us to do. So the null hypothesis is that the mean is zero. The alternative is that it is not equal to zero. So this is a situation where, if we're looking at the T-distribution right over here, it's my quick drawing of a T-distribution, if this is the mean of our T-distribution, what we care about is things that are at least 2.75 above the mean and at least 2.75 below the mean, because we care about things that are different from the mean, not just things that are greater than the mean or less than the mean. So we would look at, we would say, well, what's the probability of getting a T-value that is 2.75 or more above the mean? | Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 |
So the null hypothesis is that the mean is zero. The alternative is that it is not equal to zero. So this is a situation where, if we're looking at the T-distribution right over here, it's my quick drawing of a T-distribution, if this is the mean of our T-distribution, what we care about is things that are at least 2.75 above the mean and at least 2.75 below the mean, because we care about things that are different from the mean, not just things that are greater than the mean or less than the mean. So we would look at, we would say, well, what's the probability of getting a T-value that is 2.75 or more above the mean? And similarly, what's the probability of getting a T-value that is 2.75 or less below, or 2.75 or more below the mean? So this is negative 2.75 right over there. So what we have here is a T-table, and a T-table is a little bit different than a Z-table because there's several things going on. | Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 |
So we would look at, we would say, well, what's the probability of getting a T-value that is 2.75 or more above the mean? And similarly, what's the probability of getting a T-value that is 2.75 or less below, or 2.75 or more below the mean? So this is negative 2.75 right over there. So what we have here is a T-table, and a T-table is a little bit different than a Z-table because there's several things going on. First of all, you have your degrees of freedom. That's just going to be your sample size minus one. So in this example, our sample size is six, so six minus one is five. | Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 |
So what we have here is a T-table, and a T-table is a little bit different than a Z-table because there's several things going on. First of all, you have your degrees of freedom. That's just going to be your sample size minus one. So in this example, our sample size is six, so six minus one is five. And so we are going to be, we are going to be in this row right over here. And then what you wanna do is you wanna look up your T-value. This is T distribution critical value. | Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 |
So in this example, our sample size is six, so six minus one is five. And so we are going to be, we are going to be in this row right over here. And then what you wanna do is you wanna look up your T-value. This is T distribution critical value. So we wanna look up 2.75 on this row. And we see 2.75, it's a little bit less than that, but that's the closest value. It's a good bit more than this right over here, but it's, so it's a little bit closer to this value than this value. | Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 |
This is T distribution critical value. So we wanna look up 2.75 on this row. And we see 2.75, it's a little bit less than that, but that's the closest value. It's a good bit more than this right over here, but it's, so it's a little bit closer to this value than this value. And so our tail probability, and remember, this is only giving us this probability right over here. Our tail probability is going to be between 0.025 and 0.02, and it's going to be closer than to this one. It's gonna be approximately this. | Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 |
It's a good bit more than this right over here, but it's, so it's a little bit closer to this value than this value. And so our tail probability, and remember, this is only giving us this probability right over here. Our tail probability is going to be between 0.025 and 0.02, and it's going to be closer than to this one. It's gonna be approximately this. It'll actually be a little bit greater, because we're gonna go a little bit in that direction, because we are less than 2.757. And so we could say this is approximately 0.02. Well, if that's 0.02 approximately, the T distribution's symmetric, this is going to be approximately 0.02. | Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 |
It's gonna be approximately this. It'll actually be a little bit greater, because we're gonna go a little bit in that direction, because we are less than 2.757. And so we could say this is approximately 0.02. Well, if that's 0.02 approximately, the T distribution's symmetric, this is going to be approximately 0.02. And so our P-value, which is going to be the probability of getting a T-value that is at least 2.75 above the mean and, or 2.75 below the mean, the P-value, P-value, is going to be approximately the sum of these areas, which is 0.04. And then, of course, Katarina would wanna compare that to her significance level that she set ahead of time. And if this is lower than that, then she would reject the null hypothesis, and that would suggest the alternative. | Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 |
Sports utility vehicles, also known as SUVs, make up 12% of the vehicles she registers. Let V be the number of vehicles Amelia registers in a day until she first registers an SUV. Assume that the type of each vehicle is independent. Find the probability that Amelia registers more than four vehicles before she registers an SUV. Let's first think about what this random variable V is. It's the number of vehicles Amelia registers in a day until she registers an SUV. For example, if the first person who walks in the line or through the door has an SUV and they're trying to register it, then V would be equal to one. | Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 |
Find the probability that Amelia registers more than four vehicles before she registers an SUV. Let's first think about what this random variable V is. It's the number of vehicles Amelia registers in a day until she registers an SUV. For example, if the first person who walks in the line or through the door has an SUV and they're trying to register it, then V would be equal to one. If the first person isn't an SUV but the second person is, then V would be equal to two, so forth and so on. So this right over here is a classic geometric random variable right over here. I'll say geometric random variable. | Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 |
For example, if the first person who walks in the line or through the door has an SUV and they're trying to register it, then V would be equal to one. If the first person isn't an SUV but the second person is, then V would be equal to two, so forth and so on. So this right over here is a classic geometric random variable right over here. I'll say geometric random variable. We have a very clear success metric for each trial. Do we have an SUV or not? Each trial is independent. | Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 |
I'll say geometric random variable. We have a very clear success metric for each trial. Do we have an SUV or not? Each trial is independent. They tell us that. They are independent. The probability of success in each trial is constant. | Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 |
Each trial is independent. They tell us that. They are independent. The probability of success in each trial is constant. We have a 12% success for each new person who comes through the line. The reason why this is not a binomial random variable is that we do not have a finite number of trials. Here, we're going to keep performing trials. | Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 |
The probability of success in each trial is constant. We have a 12% success for each new person who comes through the line. The reason why this is not a binomial random variable is that we do not have a finite number of trials. Here, we're going to keep performing trials. We're going to keep serving people in the line until we get an SUV. What we have over here, when they say find the probability that Amelia registers more than four vehicles before she registers an SUV, this is the probability that V is greater than four. I encourage you, like always, pause this video and see if you can work through it. | Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 |
Here, we're going to keep performing trials. We're going to keep serving people in the line until we get an SUV. What we have over here, when they say find the probability that Amelia registers more than four vehicles before she registers an SUV, this is the probability that V is greater than four. I encourage you, like always, pause this video and see if you can work through it. We're going to assume that she's just not going to leave her desk or wherever the things are being registered. She's not going to leave the counter until someone shows up registering an SUV. We will just keep looking at people, I guess we could say, over multiple days, forever. | Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 |
I encourage you, like always, pause this video and see if you can work through it. We're going to assume that she's just not going to leave her desk or wherever the things are being registered. She's not going to leave the counter until someone shows up registering an SUV. We will just keep looking at people, I guess we could say, over multiple days, forever. It will work for an infinite number of years, just for the sake of this problem, until an SUV actually shows up. Try to figure this out. I'm assuming you've had a go. | Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 |
We will just keep looking at people, I guess we could say, over multiple days, forever. It will work for an infinite number of years, just for the sake of this problem, until an SUV actually shows up. Try to figure this out. I'm assuming you've had a go. Some of you might say, well, isn't this going to be equal to the probability that V is equal to five plus the probability that V is equal to six plus the probability that V is equal to seven, and it just goes on and on and on forever. This is actually true. You say, well, how do I calculate this? | Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 |
I'm assuming you've had a go. Some of you might say, well, isn't this going to be equal to the probability that V is equal to five plus the probability that V is equal to six plus the probability that V is equal to seven, and it just goes on and on and on forever. This is actually true. You say, well, how do I calculate this? It's just summing up an infinite number of things. Now, the key realization here is that one way to think about the probability that V is greater than four is this is the same thing as the probability that V is not less than or equal to four. These two things are equivalent. | Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 |
You say, well, how do I calculate this? It's just summing up an infinite number of things. Now, the key realization here is that one way to think about the probability that V is greater than four is this is the same thing as the probability that V is not less than or equal to four. These two things are equivalent. What's the probability that V is not less than or equal to four? This might be a slightly easier thing for you to calculate. Once again, pause the video and see if you can figure it out. | Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 |
These two things are equivalent. What's the probability that V is not less than or equal to four? This might be a slightly easier thing for you to calculate. Once again, pause the video and see if you can figure it out. What's the probability that V is not less than or equal to four? That's the same thing as the probability of first four customers or first four, I guess, people, first four, I'll say, customers, or I'll say first four cars, the customers' cars, not SUVs. This one is feeling pretty straightforward. | Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 |
Once again, pause the video and see if you can figure it out. What's the probability that V is not less than or equal to four? That's the same thing as the probability of first four customers or first four, I guess, people, first four, I'll say, customers, or I'll say first four cars, the customers' cars, not SUVs. This one is feeling pretty straightforward. What's the probability that for each customer she goes to that they're not an SUV? That's one minus 12% or 88% or 0.88. If we want to know the probability that the first four cars are not SUVs, that's 0.88 to the fourth power. | Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 |
This one is feeling pretty straightforward. What's the probability that for each customer she goes to that they're not an SUV? That's one minus 12% or 88% or 0.88. If we want to know the probability that the first four cars are not SUVs, that's 0.88 to the fourth power. That's all we have to calculate. Let's get our calculator out. I'm going to get 0.88, and I'm going to raise it to the fourth power. | Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 |
If we want to know the probability that the first four cars are not SUVs, that's 0.88 to the fourth power. That's all we have to calculate. Let's get our calculator out. I'm going to get 0.88, and I'm going to raise it to the fourth power. I'm just going to round it to the nearest. Let's see. Do they tell me to round it? | Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 |
I'm going to get 0.88, and I'm going to raise it to the fourth power. I'm just going to round it to the nearest. Let's see. Do they tell me to round it? Okay, I'll just round it to the nearest, I guess, well, hundredth. I'll just write it as 0.5997 is equal to or approximately equal to 0.5997. If you wanted to write this as a percentage, it would be approximately 59.97%. | Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 |
So let's just review factorial a little bit. So if I were to say n factorial, that of course is going to be n times n minus, sorry, times n minus one times n minus two, n minus two, and I would just keep going down until I go to times one. So I would keep decrementing n until I get to one, and then I would multiply all of those things together. So for example, in all of this is review, if I were to say three factorial, that's going to be three times two times one. If I were to say two factorial, that's going to be two times one. One factorial, by that logic, well, I just keep decrementing until I get to one, but I don't even have to decrement here. I'm already at one, so I just multiply it one. | Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
So for example, in all of this is review, if I were to say three factorial, that's going to be three times two times one. If I were to say two factorial, that's going to be two times one. One factorial, by that logic, well, I just keep decrementing until I get to one, but I don't even have to decrement here. I'm already at one, so I just multiply it one. Now what about zero factorial? This is interesting. Zero factorial. | Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
I'm already at one, so I just multiply it one. Now what about zero factorial? This is interesting. Zero factorial. So one logical thing to say, well, hey, maybe zero factorial is zero. I'm just starting with itself. It's already below one. | Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
Zero factorial. So one logical thing to say, well, hey, maybe zero factorial is zero. I'm just starting with itself. It's already below one. Maybe it is zero. Now what we will see is that this is not the case, that mathematicians have decided. And remember, this is what's interesting. | Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
It's already below one. Maybe it is zero. Now what we will see is that this is not the case, that mathematicians have decided. And remember, this is what's interesting. The factorial operation, this is something that humans have invented, that they think is just an interesting thing. It's a useful notation. So they can define what it does. | Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
And remember, this is what's interesting. The factorial operation, this is something that humans have invented, that they think is just an interesting thing. It's a useful notation. So they can define what it does. And mathematicians have found it far more useful to define zero factorial as something else. To define zero factorial as, and there's a little bit of a drum roll here, they believe zero factorial should be one. And I know, based on the reasoning, the conceptual reasoning of this, this doesn't make any sense. | Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
So they can define what it does. And mathematicians have found it far more useful to define zero factorial as something else. To define zero factorial as, and there's a little bit of a drum roll here, they believe zero factorial should be one. And I know, based on the reasoning, the conceptual reasoning of this, this doesn't make any sense. But since we've already been exposed a little bit to permutations, I'll show you why this is a useful concept, especially in the world of permutations and combinations, which is, frankly, where factorial shows up the most. I would say that most of the cases that I've ever seen, factorial in anything, has been in the situations of permutations and combinations. And in a few other things, but mainly permutations and combinations. | Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
And I know, based on the reasoning, the conceptual reasoning of this, this doesn't make any sense. But since we've already been exposed a little bit to permutations, I'll show you why this is a useful concept, especially in the world of permutations and combinations, which is, frankly, where factorial shows up the most. I would say that most of the cases that I've ever seen, factorial in anything, has been in the situations of permutations and combinations. And in a few other things, but mainly permutations and combinations. So let's review a little bit. We've said that, hey, you know, if we have n things and we want to figure out the number of ways to permute them into k spaces, it's going to be, it's going to be n factorial over n minus, over n minus k factorial. Now we've also said that if we had, if we had n things that we want to permute into n places, well this really should just be n factorial. | Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
And in a few other things, but mainly permutations and combinations. So let's review a little bit. We've said that, hey, you know, if we have n things and we want to figure out the number of ways to permute them into k spaces, it's going to be, it's going to be n factorial over n minus, over n minus k factorial. Now we've also said that if we had, if we had n things that we want to permute into n places, well this really should just be n factorial. The first place has, you know, let's just do this. So this is the first place, this is the second place, this is the third place, all the way you get to the nth place. Well, there would be n possibilities for who's in the first position or which object is in the first position. | Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
Now we've also said that if we had, if we had n things that we want to permute into n places, well this really should just be n factorial. The first place has, you know, let's just do this. So this is the first place, this is the second place, this is the third place, all the way you get to the nth place. Well, there would be n possibilities for who's in the first position or which object is in the first position. And then for each of those possibilities, there would be n minus one possibilities for which object you choose to put in the second position because you've already put one into that position. Now for each of these n times n minus one possibilities where you've placed two things, there would be n minus two possibilities of what goes in the third position and then you would just go all the way down to one. And this thing right over here is exactly what we wrote over here. | Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
Well, there would be n possibilities for who's in the first position or which object is in the first position. And then for each of those possibilities, there would be n minus one possibilities for which object you choose to put in the second position because you've already put one into that position. Now for each of these n times n minus one possibilities where you've placed two things, there would be n minus two possibilities of what goes in the third position and then you would just go all the way down to one. And this thing right over here is exactly what we wrote over here. This is equal to, this is equal to n factorial. But if we directly applied this formula, if we applied this formula, this would need to be n factorial over n minus n factorial. And then you might see why this is interesting because this is going to be n factorial over zero factorial. | Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
And this thing right over here is exactly what we wrote over here. This is equal to, this is equal to n factorial. But if we directly applied this formula, if we applied this formula, this would need to be n factorial over n minus n factorial. And then you might see why this is interesting because this is going to be n factorial over zero factorial. So in order for this formula to apply for even in the case where k is equal to n, even in the case where k is equal to n, which is this one right over here, and for that to be consistent with just plain old logic, zero factorial needs to be equal to one. And so the mathematics community has decided, hey, this thing we've constructed called factorial, you know, we've said, hey, you put an exclamation mark behind something. In all of our heads, we say you kind of count down that number all the way to one and you keep multiplying them. | Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
And then you might see why this is interesting because this is going to be n factorial over zero factorial. So in order for this formula to apply for even in the case where k is equal to n, even in the case where k is equal to n, which is this one right over here, and for that to be consistent with just plain old logic, zero factorial needs to be equal to one. And so the mathematics community has decided, hey, this thing we've constructed called factorial, you know, we've said, hey, you put an exclamation mark behind something. In all of our heads, we say you kind of count down that number all the way to one and you keep multiplying them. For zero, we're just gonna define this. We're just gonna define, make a mathematical definition. We're just gonna say zero factorial is equal to one. | Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
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