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Hopefully that makes a little sense of how this is going to seem. My phone is ringing. No, I'll ignore that. Because the normal distribution is so important. And actually, my 9-week-old son is watching, so this is the first time I have a live audience. But I thought he might pick up something about the normal distribution. But anyway, the blue line right here, and I'll trace it maybe in yellow just so you can see it, is the plot of the binomial distribution. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
Because the normal distribution is so important. And actually, my 9-week-old son is watching, so this is the first time I have a live audience. But I thought he might pick up something about the normal distribution. But anyway, the blue line right here, and I'll trace it maybe in yellow just so you can see it, is the plot of the binomial distribution. And I connected the lines, but you see the binomial distribution look something more like this. Where this is the probability of getting to minus 4. This is the probability of going to minus 2. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
But anyway, the blue line right here, and I'll trace it maybe in yellow just so you can see it, is the plot of the binomial distribution. And I connected the lines, but you see the binomial distribution look something more like this. Where this is the probability of getting to minus 4. This is the probability of going to minus 2. This right here is the probability of ending up nowhere. And then this is the probability. Actually, no, the point is right here. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
This is the probability of going to minus 2. This right here is the probability of ending up nowhere. And then this is the probability. Actually, no, the point is right here. This is the probability of ending up 2 to the right. And this is the probability of ending 4 to the right. This is the binomial distribution. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
Actually, no, the point is right here. This is the probability of ending up 2 to the right. And this is the probability of ending 4 to the right. This is the binomial distribution. I just plotted these points right here. This is 0.375. This is 0.375. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
This is the binomial distribution. I just plotted these points right here. This is 0.375. This is 0.375. That's the height of that. Now, what I wanted to show you is that the normal distribution approximates the binomial distribution. So this right here, I wanted to say, what does the normal distribution tell me is the probability of ending up with exactly 0 left steps. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
This is 0.375. That's the height of that. Now, what I wanted to show you is that the normal distribution approximates the binomial distribution. So this right here, I wanted to say, what does the normal distribution tell me is the probability of ending up with exactly 0 left steps. And then this is a little bit tricky because the binomial distribution is a discrete probability distribution. You could just look at this chart or look here and you say, what is the probability of having exactly, let's say, 1 left step and 3 right steps, which puts me right here. Well, you just look at this chart and you say, oh, that puts me right there. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So this right here, I wanted to say, what does the normal distribution tell me is the probability of ending up with exactly 0 left steps. And then this is a little bit tricky because the binomial distribution is a discrete probability distribution. You could just look at this chart or look here and you say, what is the probability of having exactly, let's say, 1 left step and 3 right steps, which puts me right here. Well, you just look at this chart and you say, oh, that puts me right there. I just read that probability. It's actually 0.25. And I say, oh, I have a 25% chance of ending up 2 steps to the right. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
Well, you just look at this chart and you say, oh, that puts me right there. I just read that probability. It's actually 0.25. And I say, oh, I have a 25% chance of ending up 2 steps to the right. There's a 25% chance. The normal distribution function is a continuous probability distribution. So it's a continuous curve. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And I say, oh, I have a 25% chance of ending up 2 steps to the right. There's a 25% chance. The normal distribution function is a continuous probability distribution. So it's a continuous curve. It looks like that. It's the bell curve. And it goes off to infinity and starts approaching 0 on both sides. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So it's a continuous curve. It looks like that. It's the bell curve. And it goes off to infinity and starts approaching 0 on both sides. It looks something like that. But this is a continuous probability distribution. You can't just take a point here and you say, what's the probability that I end up 2 feet to the right? | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And it goes off to infinity and starts approaching 0 on both sides. It looks something like that. But this is a continuous probability distribution. You can't just take a point here and you say, what's the probability that I end up 2 feet to the right? Because if you just say that, the actual probability of being exactly, and you should watch my video on probability density functions, but the probability of being exactly 2 feet to the right, exactly, I mean, I'm talking to the atom, is close to 0. So you actually have to specify a range around this. And what I assume in this is I assume that within, essentially, a half a foot in either direction, if we're talking about feet. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
You can't just take a point here and you say, what's the probability that I end up 2 feet to the right? Because if you just say that, the actual probability of being exactly, and you should watch my video on probability density functions, but the probability of being exactly 2 feet to the right, exactly, I mean, I'm talking to the atom, is close to 0. So you actually have to specify a range around this. And what I assume in this is I assume that within, essentially, a half a foot in either direction, if we're talking about feet. So to figure that out, what I did here is I took the value of the probability density function there, and I'll show you how I evaluated that. And then I multiplied that by 1. So it gives me this area. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And what I assume in this is I assume that within, essentially, a half a foot in either direction, if we're talking about feet. So to figure that out, what I did here is I took the value of the probability density function there, and I'll show you how I evaluated that. And then I multiplied that by 1. So it gives me this area. And I use that as an approximation for this area. If you really wanted to be particular about it, what you would do is you would take the integral of this curve between this point and this point as a better approximation. We'll do that in the future. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So it gives me this area. And I use that as an approximation for this area. If you really wanted to be particular about it, what you would do is you would take the integral of this curve between this point and this point as a better approximation. We'll do that in the future. But right now, I just want to give you the intuition that the binomial distribution really does converge to the normal distribution. So what is this number right here? Well, I said, what is the probability? | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
We'll do that in the future. But right now, I just want to give you the intuition that the binomial distribution really does converge to the normal distribution. So what is this number right here? Well, I said, what is the probability? Well, let me do something like that. Let's say this one right here, because I don't want to use the 0. So what is the probability that I take one left step? | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
Well, I said, what is the probability? Well, let me do something like that. Let's say this one right here, because I don't want to use the 0. So what is the probability that I take one left step? I kind of used left steps as a success. So what is the probability of 1? And that equaled 1 over the standard deviation. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So what is the probability that I take one left step? I kind of used left steps as a success. So what is the probability of 1? And that equaled 1 over the standard deviation. When I only took 4 steps, the standard deviation was 1. So 1 over 1. Actually, let me change this, because I think it might be let me change it to a higher number. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And that equaled 1 over the standard deviation. When I only took 4 steps, the standard deviation was 1. So 1 over 1. Actually, let me change this, because I think it might be let me change it to a higher number. Let's see, if I make this, I don't know. Let me go back to the example where I'm at 10. All right, so if this is at 10, let me go back to my drawing tool. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
Actually, let me change this, because I think it might be let me change it to a higher number. Let's see, if I make this, I don't know. Let me go back to the example where I'm at 10. All right, so if this is at 10, let me go back to my drawing tool. So this calculation right here, let me do this calculation. Actually, even better, let me do this calculation right here. All right, so what's the probability that I have 2 left steps, and if I have 2 left steps, I took a total of 10 steps, so I'm going to have 8 right steps, and that puts me 6 to the right. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
All right, so if this is at 10, let me go back to my drawing tool. So this calculation right here, let me do this calculation. Actually, even better, let me do this calculation right here. All right, so what's the probability that I have 2 left steps, and if I have 2 left steps, I took a total of 10 steps, so I'm going to have 8 right steps, and that puts me 6 to the right. So that's this point right here. So what's that probability? How do I figure this out using the probability density function? | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
All right, so what's the probability that I have 2 left steps, and if I have 2 left steps, I took a total of 10 steps, so I'm going to have 8 right steps, and that puts me 6 to the right. So that's this point right here. So what's that probability? How do I figure this out using the probability density function? How do I figure this out? Well, I say the probability of taking 2 left steps, that's how I calculated it. If you actually click on the cell, you'll see that, is equal to 1 over the standard deviation, 1.581. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
How do I figure this out using the probability density function? How do I figure this out? Well, I say the probability of taking 2 left steps, that's how I calculated it. If you actually click on the cell, you'll see that, is equal to 1 over the standard deviation, 1.581. And I just directly referenced the cell there. Divide it times the square root of 2 pi. And I always go on in awe of the whole notion of e to the i pi is equal to negative 1 and all of that. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
If you actually click on the cell, you'll see that, is equal to 1 over the standard deviation, 1.581. And I just directly referenced the cell there. Divide it times the square root of 2 pi. And I always go on in awe of the whole notion of e to the i pi is equal to negative 1 and all of that. But this is another amazing thing, that all of a sudden we have this, as we take many, many, many trials, we have this formula that has e and pi in it and square roots. But once again, these two numbers just keep showing up, and it tells you something about the order of the universe with a capital O. But let's see, so times e to the minus 1 half times x. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And I always go on in awe of the whole notion of e to the i pi is equal to negative 1 and all of that. But this is another amazing thing, that all of a sudden we have this, as we take many, many, many trials, we have this formula that has e and pi in it and square roots. But once again, these two numbers just keep showing up, and it tells you something about the order of the universe with a capital O. But let's see, so times e to the minus 1 half times x. Well, the x is what we're trying to calculate. 2 successes. So to have exactly 2 left, so it's 2 minus the mean. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
But let's see, so times e to the minus 1 half times x. Well, the x is what we're trying to calculate. 2 successes. So to have exactly 2 left, so it's 2 minus the mean. So the mean is 5. Divided by the standard deviation, divided by 1.581. All of that squared. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So to have exactly 2 left, so it's 2 minus the mean. So the mean is 5. Divided by the standard deviation, divided by 1.581. All of that squared. That's where this calculation came from. And so I told you in the last one, I mean, this just tells you, this right here just tells me this value up here. And I assume that the probability, if I want to know this exact probability, it's the area of this. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
All of that squared. That's where this calculation came from. And so I told you in the last one, I mean, this just tells you, this right here just tells me this value up here. And I assume that the probability, if I want to know this exact probability, it's the area of this. And if I just take one line, the area is 0. So to be exactly 2 feet away using the, remember, I mean, in this case, you can only be 2 feet away because we're taking very exact steps. But what the normal distribution is, it's a continuous probability density function. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And I assume that the probability, if I want to know this exact probability, it's the area of this. And if I just take one line, the area is 0. So to be exactly 2 feet away using the, remember, I mean, in this case, you can only be 2 feet away because we're taking very exact steps. But what the normal distribution is, it's a continuous probability density function. So it can tell us, what's the probability of being 2.183 feet away? Which obviously can only happen if we're kind of taking infinitely small steps every time. But that's what its use is. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
But what the normal distribution is, it's a continuous probability density function. So it can tell us, what's the probability of being 2.183 feet away? Which obviously can only happen if we're kind of taking infinitely small steps every time. But that's what its use is. It happens kind of when you start taking an infinite number of steps. But it can approximate the discrete. And the way I approximate it is I say, oh, what's the probability of being within a foot of that? | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
But that's what its use is. It happens kind of when you start taking an infinite number of steps. But it can approximate the discrete. And the way I approximate it is I say, oh, what's the probability of being within a foot of that? And so I multiply this height, which I calculate here, times 1. So let's say this has a base of 1 to calculate this area, which I use as an approximation. So you just multiply that times 1, and that's what you get here. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And the way I approximate it is I say, oh, what's the probability of being within a foot of that? And so I multiply this height, which I calculate here, times 1. So let's say this has a base of 1 to calculate this area, which I use as an approximation. So you just multiply that times 1, and that's what you get here. And I just want to show you, I mean, even with just 10 trials, the curves for the normal distribution here is in purple, and the binomial distribution is in blue. So they're almost right on top of each other. I mean, when the number of steps I took was a little bit smaller, and as you take many, many, many, many more steps, they almost converge right on top of each other. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So you just multiply that times 1, and that's what you get here. And I just want to show you, I mean, even with just 10 trials, the curves for the normal distribution here is in purple, and the binomial distribution is in blue. So they're almost right on top of each other. I mean, when the number of steps I took was a little bit smaller, and as you take many, many, many, many more steps, they almost converge right on top of each other. And I encourage you to play with this spreadsheet. And actually, let me show you that they converge. So I made another, there's a convergence worksheet on this spreadsheet as well, if you click on the bottom tab on convergence. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
I mean, when the number of steps I took was a little bit smaller, and as you take many, many, many, many more steps, they almost converge right on top of each other. And I encourage you to play with this spreadsheet. And actually, let me show you that they converge. So I made another, there's a convergence worksheet on this spreadsheet as well, if you click on the bottom tab on convergence. And I did, this is the same thing, but I just wanted to show you what happens at any given point. So let's say that I wanted to, let me explain this part, this spreadsheet to you. So this is, what's the probability of moving left, right, this is, so this is just saying, I'm just fixing a point, what's the probability, and you could change this, of my final position being 10. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So I made another, there's a convergence worksheet on this spreadsheet as well, if you click on the bottom tab on convergence. And I did, this is the same thing, but I just wanted to show you what happens at any given point. So let's say that I wanted to, let me explain this part, this spreadsheet to you. So this is, what's the probability of moving left, right, this is, so this is just saying, I'm just fixing a point, what's the probability, and you could change this, of my final position being 10. And then this essentially tells you that if I take 10 moves, then for my final position to be 10 to the right, I have to take 10 right moves and 0 left moves. That's a typo right there, it should be moves, not moves-ed. If I take 20 moves to end up 10 moves to the right, then I have to make 15 right moves and 5 left moves. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So this is, what's the probability of moving left, right, this is, so this is just saying, I'm just fixing a point, what's the probability, and you could change this, of my final position being 10. And then this essentially tells you that if I take 10 moves, then for my final position to be 10 to the right, I have to take 10 right moves and 0 left moves. That's a typo right there, it should be moves, not moves-ed. If I take 20 moves to end up 10 moves to the right, then I have to make 15 right moves and 5 left moves. And likewise, if I take a total of 80 moves, if I take 80 flips of my coin to make me go left to right, in order to end up 10 to the right, I have to take 45 right moves and 35 left moves, in any order. And it'll end up with 10 to the right. So what I want to figure out is, as I start taking a bunch of total moves, it's like my total moves, I mean here I max it out at 170, but you could kind of say, if I started flipping this coin an infinite number of times, I want to figure out what's the probability that my final position is 10 to the right. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
If I take 20 moves to end up 10 moves to the right, then I have to make 15 right moves and 5 left moves. And likewise, if I take a total of 80 moves, if I take 80 flips of my coin to make me go left to right, in order to end up 10 to the right, I have to take 45 right moves and 35 left moves, in any order. And it'll end up with 10 to the right. So what I want to figure out is, as I start taking a bunch of total moves, it's like my total moves, I mean here I max it out at 170, but you could kind of say, if I started flipping this coin an infinite number of times, I want to figure out what's the probability that my final position is 10 to the right. And what I want to show you is that as you take more and more moves, the normal distribution becomes a better and better approximation for the binomial distribution. So right here, this calculates the binomial probability, just the way we did it before. And you could look at the cell to figure it out. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So what I want to figure out is, as I start taking a bunch of total moves, it's like my total moves, I mean here I max it out at 170, but you could kind of say, if I started flipping this coin an infinite number of times, I want to figure out what's the probability that my final position is 10 to the right. And what I want to show you is that as you take more and more moves, the normal distribution becomes a better and better approximation for the binomial distribution. So right here, this calculates the binomial probability, just the way we did it before. And you could look at the cell to figure it out. You know, this says 10, you know, I use left moves as a success. So this is 10 choose 0, and we know what that is, it's 10 factorial over 0 factorial over 10 minus 0 factorial, times 0.5 to the 0, times 0.5 to the 10. That's where that number comes from. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And you could look at the cell to figure it out. You know, this says 10, you know, I use left moves as a success. So this is 10 choose 0, and we know what that is, it's 10 factorial over 0 factorial over 10 minus 0 factorial, times 0.5 to the 0, times 0.5 to the 10. That's where that number comes from. If I go to, let's say, this one right here, this one right here is calculated. Actually, let me write it out, because I think it's interesting. I have a total of 60 total moves, so it's 60 factorial over, I have to have 25 left moves, so 25 factorial, 60 minus 25 factorial, times the probability of a left move, and I have 25 of them, times the probability of a right move, and I have 35 of those. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
That's where that number comes from. If I go to, let's say, this one right here, this one right here is calculated. Actually, let me write it out, because I think it's interesting. I have a total of 60 total moves, so it's 60 factorial over, I have to have 25 left moves, so 25 factorial, 60 minus 25 factorial, times the probability of a left move, and I have 25 of them, times the probability of a right move, and I have 35 of those. So that's just what the binomial probability distribution will tell us. And then it figures out the mean and the variance for each of those circumstances, and you could look at the formula, but the mean is just the probability of having a left move times the total number of moves. The variance is probability of left times probability of right times total number of moves. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
I have a total of 60 total moves, so it's 60 factorial over, I have to have 25 left moves, so 25 factorial, 60 minus 25 factorial, times the probability of a left move, and I have 25 of them, times the probability of a right move, and I have 35 of those. So that's just what the binomial probability distribution will tell us. And then it figures out the mean and the variance for each of those circumstances, and you could look at the formula, but the mean is just the probability of having a left move times the total number of moves. The variance is probability of left times probability of right times total number of moves. And then the normal probability, once again, I just use the normal probability, I just use the, so I approximate it the same way. So for example, in this situation right here, and Excel has a normal distribution function, but I actually typed in the formula because I wanted to kind of see what was under the covers for that function that Excel actually has. So I actually say, what's the probability of 45 left moves? | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
The variance is probability of left times probability of right times total number of moves. And then the normal probability, once again, I just use the normal probability, I just use the, so I approximate it the same way. So for example, in this situation right here, and Excel has a normal distribution function, but I actually typed in the formula because I wanted to kind of see what was under the covers for that function that Excel actually has. So I actually say, what's the probability of 45 left moves? So I say the probability of 45 left moves is equal to 1 over the standard deviation. So in this situation, the standard deviation is the square root of 25, so it's 5 times 2 pi times e to the minus 1 half times 45 minus the mean, minus 50, over the standard deviation, which we figured out was 5, squared. So that tells me the value of what the normal distribution tells me for this situation with this standard deviation and this mean. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So I actually say, what's the probability of 45 left moves? So I say the probability of 45 left moves is equal to 1 over the standard deviation. So in this situation, the standard deviation is the square root of 25, so it's 5 times 2 pi times e to the minus 1 half times 45 minus the mean, minus 50, over the standard deviation, which we figured out was 5, squared. So that tells me the value of what the normal distribution tells me for this situation with this standard deviation and this mean. And then I multiply that by 1, so you don't see that in formula, I don't actually write times 1, to actually figure out the area under the curve, right? Because remember, it's a continuous distribution function like that. This right here just gives me the value. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So that tells me the value of what the normal distribution tells me for this situation with this standard deviation and this mean. And then I multiply that by 1, so you don't see that in formula, I don't actually write times 1, to actually figure out the area under the curve, right? Because remember, it's a continuous distribution function like that. This right here just gives me the value. But to figure out the probability of being within a foot of it, I have to multiply it by 1. Or I'm approximating, really. I really should take the integral from there to there, but this little rectangle is a pretty good approximation. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
This right here just gives me the value. But to figure out the probability of being within a foot of it, I have to multiply it by 1. Or I'm approximating, really. I really should take the integral from there to there, but this little rectangle is a pretty good approximation. And this chart, I show you that as the total number of moves gets larger and larger, the difference between what the normal probability distribution tells us and the binomial probability distribution tells us gets smaller and smaller in terms of the probability of you ending up 10 moves to the right. And you can change this number here. Let me change it just to show you. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
I really should take the integral from there to there, but this little rectangle is a pretty good approximation. And this chart, I show you that as the total number of moves gets larger and larger, the difference between what the normal probability distribution tells us and the binomial probability distribution tells us gets smaller and smaller in terms of the probability of you ending up 10 moves to the right. And you can change this number here. Let me change it just to show you. You could say, what's the probability of being 15 moves to the right? And actually, no, that one doesn't. 15 moves to the right. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
Let me change it just to show you. You could say, what's the probability of being 15 moves to the right? And actually, no, that one doesn't. 15 moves to the right. It looks like it kind of, no, that's not. Let's see, 10. And if I go 12, it converges. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
15 moves to the right. It looks like it kind of, no, that's not. Let's see, 10. And if I go 12, it converges. And then if you go to 13, I think that something's happening with the floating point error. Because when you get to large factorials, I think something weird happens out here. But like if you do 3, something weird is happening. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And if I go 12, it converges. And then if you go to 13, I think that something's happening with the floating point error. Because when you get to large factorials, I think something weird happens out here. But like if you do 3, something weird is happening. 5, 10. Yeah, you maybe have to just get even further out. So for 10, you can see clearly that it converges. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
But like if you do 3, something weird is happening. 5, 10. Yeah, you maybe have to just get even further out. So for 10, you can see clearly that it converges. And I'll try to figure out why I was getting those weird sawtooth patterns. If you get 11, no, everything is, maybe while I do screen capture, something weird is happening. But anyway, the whole point of this was to show you that if you wanted to figure out the probability of being 10 moves to the right, as you take more and more flips of your coin, the normal distribution becomes a much better approximation for the actual binomial distribution. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So for 10, you can see clearly that it converges. And I'll try to figure out why I was getting those weird sawtooth patterns. If you get 11, no, everything is, maybe while I do screen capture, something weird is happening. But anyway, the whole point of this was to show you that if you wanted to figure out the probability of being 10 moves to the right, as you take more and more flips of your coin, the normal distribution becomes a much better approximation for the actual binomial distribution. And as you approach infinity, they actually converge to each other. Anyway, that's all for this video. I'll actually do several more videos on the normal distribution, just because it is such an important concept. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
Probability, a word that you've probably heard a lot of and you are probably a little bit familiar with it, but hopefully this will give you a little deeper understanding. So let's say that I have a fair coin over here. So when I talk about a fair coin, I mean that it has an equal chance of landing on one side or another. So you can maybe view it as the sides are equal, the weight is the same on either side. If I flip it in the air, it's not more likely to land on one side or the other. It's equally likely. And so you have one side of this coin, so this would be the heads, I guess. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
So you can maybe view it as the sides are equal, the weight is the same on either side. If I flip it in the air, it's not more likely to land on one side or the other. It's equally likely. And so you have one side of this coin, so this would be the heads, I guess. Try to draw George Washington. I'll assume it's a quarter of some kind. And then the other side, of course, is the tails. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
And so you have one side of this coin, so this would be the heads, I guess. Try to draw George Washington. I'll assume it's a quarter of some kind. And then the other side, of course, is the tails. So that is heads. The other side right over there is tails. And so if I were to ask you, what is the probability? | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
And then the other side, of course, is the tails. So that is heads. The other side right over there is tails. And so if I were to ask you, what is the probability? I'm going to flip a coin, and I want to know what is the probability of getting heads. And I could write that like this. The probability of getting heads. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
And so if I were to ask you, what is the probability? I'm going to flip a coin, and I want to know what is the probability of getting heads. And I could write that like this. The probability of getting heads. And you probably, just based on that question, have a sense of what probability is asking. It's asking for some type of way of getting your hands around an event that's fundamentally random. We don't know whether it's heads or tails, but we can start to describe the chances of it being heads or tails. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
The probability of getting heads. And you probably, just based on that question, have a sense of what probability is asking. It's asking for some type of way of getting your hands around an event that's fundamentally random. We don't know whether it's heads or tails, but we can start to describe the chances of it being heads or tails. And we'll talk about different ways of describing that. So one way to think about it, and this is the way that probability tends to be introduced in textbooks, is you say, well look, how many different equally likely possibilities are there? So how many equally likely possibilities? | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
We don't know whether it's heads or tails, but we can start to describe the chances of it being heads or tails. And we'll talk about different ways of describing that. So one way to think about it, and this is the way that probability tends to be introduced in textbooks, is you say, well look, how many different equally likely possibilities are there? So how many equally likely possibilities? So number of equally likely possibilities. And of the number of equally likely possibilities, I care about the number that contain my event right here. So the number of possibilities that meet my constraint. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
So how many equally likely possibilities? So number of equally likely possibilities. And of the number of equally likely possibilities, I care about the number that contain my event right here. So the number of possibilities that meet my constraint. That meet my conditions. So in the case of the probability of figuring out heads, what is the number of equally likely possibilities? Well there's only two possibilities. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
So the number of possibilities that meet my constraint. That meet my conditions. So in the case of the probability of figuring out heads, what is the number of equally likely possibilities? Well there's only two possibilities. We're assuming that the coin can't land on its corner and just stand straight up. We're assuming that it lands flat. So there's two possibilities here. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
Well there's only two possibilities. We're assuming that the coin can't land on its corner and just stand straight up. We're assuming that it lands flat. So there's two possibilities here. Two equally likely possibilities. You could either get heads or you could get tails. And what's the number of possibilities that meet my conditions? | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
So there's two possibilities here. Two equally likely possibilities. You could either get heads or you could get tails. And what's the number of possibilities that meet my conditions? Well there's only one. The condition of heads. So it'll be 1 over 2. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
And what's the number of possibilities that meet my conditions? Well there's only one. The condition of heads. So it'll be 1 over 2. So the one way to think about it is the probability of getting heads is equal to 1 over 2. Is equal to 1 half. If I wanted to write that as a percentage, we know that 1 half is the same thing as 50%. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
So it'll be 1 over 2. So the one way to think about it is the probability of getting heads is equal to 1 over 2. Is equal to 1 half. If I wanted to write that as a percentage, we know that 1 half is the same thing as 50%. Now another way to think about or conceptualize probability that will give you this exact same answer, is to say, well if I were to run the experiment of flipping a coin. So this flip, you view this as an experiment. I know this isn't the kind of experiment that you're used to. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
If I wanted to write that as a percentage, we know that 1 half is the same thing as 50%. Now another way to think about or conceptualize probability that will give you this exact same answer, is to say, well if I were to run the experiment of flipping a coin. So this flip, you view this as an experiment. I know this isn't the kind of experiment that you're used to. You know, you normally think an experiment is doing something in chemistry or physics or all the rest. But an experiment is every time you run this random event. So one way to think about probability is if I were to do this experiment, an experiment many, many, many times. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
I know this isn't the kind of experiment that you're used to. You know, you normally think an experiment is doing something in chemistry or physics or all the rest. But an experiment is every time you run this random event. So one way to think about probability is if I were to do this experiment, an experiment many, many, many times. If I were to do it a thousand times or a million times or a billion times or a trillion times, and the more the better. What percentage of those would give me what I care about? What percentage of those would give me heads? | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
So one way to think about probability is if I were to do this experiment, an experiment many, many, many times. If I were to do it a thousand times or a million times or a billion times or a trillion times, and the more the better. What percentage of those would give me what I care about? What percentage of those would give me heads? And so another way to think about this 50% probability of getting heads, is if I were to run this experiment tons of times. If I were to run this forever and closer or an infinite number of times, what percentage of those would be heads? You would get this 50%. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
What percentage of those would give me heads? And so another way to think about this 50% probability of getting heads, is if I were to run this experiment tons of times. If I were to run this forever and closer or an infinite number of times, what percentage of those would be heads? You would get this 50%. And you can run that simulation. You can flip a coin. And it's actually a fun thing to do. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
You would get this 50%. And you can run that simulation. You can flip a coin. And it's actually a fun thing to do. I encourage you to do it. If you take 100 or 200 quarters or pennies, stick them in a big box, shake the box. So you're kind of simultaneously flipping all of the coins. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
And it's actually a fun thing to do. I encourage you to do it. If you take 100 or 200 quarters or pennies, stick them in a big box, shake the box. So you're kind of simultaneously flipping all of the coins. And then count how many of those are going to be heads. And you're going to see that the larger the number that you are doing, the more likely you're going to get something really close to 50%. There's always some chance, even if you flip the coin a million times, there's some super duper small chance that you get all tails. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
So you're kind of simultaneously flipping all of the coins. And then count how many of those are going to be heads. And you're going to see that the larger the number that you are doing, the more likely you're going to get something really close to 50%. There's always some chance, even if you flip the coin a million times, there's some super duper small chance that you get all tails. But the more you do, the more likely that things are going to trend towards 50% of them are going to be heads. Now let's just apply these same ideas. And while we're starting with probability, at least kind of the basic, this is probably an easier thing to conceptualize. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
There's always some chance, even if you flip the coin a million times, there's some super duper small chance that you get all tails. But the more you do, the more likely that things are going to trend towards 50% of them are going to be heads. Now let's just apply these same ideas. And while we're starting with probability, at least kind of the basic, this is probably an easier thing to conceptualize. But a lot of times this is actually a helpful one too, this idea that if you run the experiment many, many, many, many times, what percentage of those trials are going to give you what you're asking for? In this case, it was heads. Now let's do another very typical example when you first learn probability. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
And while we're starting with probability, at least kind of the basic, this is probably an easier thing to conceptualize. But a lot of times this is actually a helpful one too, this idea that if you run the experiment many, many, many, many times, what percentage of those trials are going to give you what you're asking for? In this case, it was heads. Now let's do another very typical example when you first learn probability. And this is the idea of rolling a die. So here's my die right over here. And of course you have the different sides of the die. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
Now let's do another very typical example when you first learn probability. And this is the idea of rolling a die. So here's my die right over here. And of course you have the different sides of the die. So that's the 1, that's the 2, that's the 3. And what I want to do, and we know of course that there are, and I'm assuming this is a fair die, and so there are 6 equally likely possibilities. When you roll this, you could get a 1, a 2, a 3, a 4, a 5, or a 6. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
And of course you have the different sides of the die. So that's the 1, that's the 2, that's the 3. And what I want to do, and we know of course that there are, and I'm assuming this is a fair die, and so there are 6 equally likely possibilities. When you roll this, you could get a 1, a 2, a 3, a 4, a 5, or a 6. And they are all equally likely. So if I were to ask you, what is the probability, given that I'm rolling a fair die, so the experiment is rolling this fair die, what is the probability of getting a 1? Well, what are the number of equally likely possibilities? | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
When you roll this, you could get a 1, a 2, a 3, a 4, a 5, or a 6. And they are all equally likely. So if I were to ask you, what is the probability, given that I'm rolling a fair die, so the experiment is rolling this fair die, what is the probability of getting a 1? Well, what are the number of equally likely possibilities? Well, I have 6 equally likely possibilities. And how many of those meet my conditions? Well, only one of them meets my condition, that right there. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
Well, what are the number of equally likely possibilities? Well, I have 6 equally likely possibilities. And how many of those meet my conditions? Well, only one of them meets my condition, that right there. So there is a 1 6 probability of rolling a 1. What is the probability of rolling a 1 or a 6? Well, once again, there are 6 equally likely possibilities for what I can get. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
Well, only one of them meets my condition, that right there. So there is a 1 6 probability of rolling a 1. What is the probability of rolling a 1 or a 6? Well, once again, there are 6 equally likely possibilities for what I can get. And there are now 2 possibilities that meet my conditions. I could roll a 1 or I could roll a 6. So now there are 2 possibilities that meet my constraints, my conditions. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
Well, once again, there are 6 equally likely possibilities for what I can get. And there are now 2 possibilities that meet my conditions. I could roll a 1 or I could roll a 6. So now there are 2 possibilities that meet my constraints, my conditions. So there is a 1 3 probability of rolling a 1 or a 6. Now what is the probability, this might seem a little silly to even ask this question, but I'll ask it just to make it clear. What is the probability of rolling a 2 and a 3? | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
So now there are 2 possibilities that meet my constraints, my conditions. So there is a 1 3 probability of rolling a 1 or a 6. Now what is the probability, this might seem a little silly to even ask this question, but I'll ask it just to make it clear. What is the probability of rolling a 2 and a 3? And I'm just talking about one roll of the die. Well, in any roll of the die, I can only get a 2 or a 3. I'm not talking about taking 2 rolls of this die. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
What is the probability of rolling a 2 and a 3? And I'm just talking about one roll of the die. Well, in any roll of the die, I can only get a 2 or a 3. I'm not talking about taking 2 rolls of this die. So in this situation, there are 6 possibilities, but none of these possibilities are 2 and a 3. 2 and a 3 cannot exist on one trial. You cannot get a 2 and a 3 in the same experiment. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
I'm not talking about taking 2 rolls of this die. So in this situation, there are 6 possibilities, but none of these possibilities are 2 and a 3. 2 and a 3 cannot exist on one trial. You cannot get a 2 and a 3 in the same experiment. Getting a 2 and a 3 are mutually exclusive events. They cannot happen at the same time. So the probability of this is actually 0. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
You cannot get a 2 and a 3 in the same experiment. Getting a 2 and a 3 are mutually exclusive events. They cannot happen at the same time. So the probability of this is actually 0. There's no way to roll this normal die and all of a sudden you get a 2 and a 3. In fact, I don't want to confuse you with that because it's kind of abstract and impossible. So let's cross this out right over here. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
So the probability of this is actually 0. There's no way to roll this normal die and all of a sudden you get a 2 and a 3. In fact, I don't want to confuse you with that because it's kind of abstract and impossible. So let's cross this out right over here. Now what is the probability of getting an even number? So once again, you have 6 equally likely possibilities when I roll that die. And which of these possibilities meet my conditions, the condition of being even? | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
So let's cross this out right over here. Now what is the probability of getting an even number? So once again, you have 6 equally likely possibilities when I roll that die. And which of these possibilities meet my conditions, the condition of being even? Well, 2 is even, 4 is even, and 6 is even. So 3 of the possibilities meet my conditions, meet my constraints. So this is 1 half. | Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3 |
So let's say we have six people. We have person A, we have person B, we have person C, person D, person E, and we have person F. So we have six people, and for the sake of this video, we're going to say, oh, we want to figure out all the scenarios, all the possibilities, all the permutations, all the ways that we could put them into three chairs. So that's chair number one, chair number two, and chair number three. This is all a review, this is covered in the permutations video, but it'll be very instructive as we move into a new concept. So what are all of the permutations of putting six different people into three chairs? Well, like we've seen before, we could start with the first chair, and we could say, look, if we haven't seated anyone yet, how many different people could we put in chair number one? Well, there's six different people who could be in chair number one. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
This is all a review, this is covered in the permutations video, but it'll be very instructive as we move into a new concept. So what are all of the permutations of putting six different people into three chairs? Well, like we've seen before, we could start with the first chair, and we could say, look, if we haven't seated anyone yet, how many different people could we put in chair number one? Well, there's six different people who could be in chair number one. Let me do that in a different color. There are six people who could be in chair number one. Six different scenarios for who sits in chair number one. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
Well, there's six different people who could be in chair number one. Let me do that in a different color. There are six people who could be in chair number one. Six different scenarios for who sits in chair number one. Now, for each of those six scenarios, how many people, how many different people could sit in chair number two? Well, in each of those six scenarios, we've taken one of the six people to sit in chair number one, so that means you have five out of the six people left to sit in chair number two. Or another way to think about it is there's six scenarios of someone in chair number one, and for each of those six, you have five scenarios for who's in chair number two. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
Six different scenarios for who sits in chair number one. Now, for each of those six scenarios, how many people, how many different people could sit in chair number two? Well, in each of those six scenarios, we've taken one of the six people to sit in chair number one, so that means you have five out of the six people left to sit in chair number two. Or another way to think about it is there's six scenarios of someone in chair number one, and for each of those six, you have five scenarios for who's in chair number two. So you have a total of 30 scenarios where you have seated six people in the first two chairs. And now, if you want to say, well, what about for the three chairs? Well, for each of these 30 scenarios, how many different people could you put in chair number three? | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
Or another way to think about it is there's six scenarios of someone in chair number one, and for each of those six, you have five scenarios for who's in chair number two. So you have a total of 30 scenarios where you have seated six people in the first two chairs. And now, if you want to say, well, what about for the three chairs? Well, for each of these 30 scenarios, how many different people could you put in chair number three? Well, you're still gonna have four people standing up, not in chairs. So for each of these 30 scenarios, you have four people who you could put in chair number three. So your total number of scenarios or your total number of permutations, where we care who's sitting in which chair, is six times five times four, which is equal to 120 permutations. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
Well, for each of these 30 scenarios, how many different people could you put in chair number three? Well, you're still gonna have four people standing up, not in chairs. So for each of these 30 scenarios, you have four people who you could put in chair number three. So your total number of scenarios or your total number of permutations, where we care who's sitting in which chair, is six times five times four, which is equal to 120 permutations. Permutations. Now, permutations. Now, it's worth thinking about what permutations are counting. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
So your total number of scenarios or your total number of permutations, where we care who's sitting in which chair, is six times five times four, which is equal to 120 permutations. Permutations. Now, permutations. Now, it's worth thinking about what permutations are counting. Now, remember, we care, when we're talking about permutations, we care about who's sitting in which chair. So, for example, this is one permutation, and this would be counted as another permutation. And this would be counted as another permutation. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
Now, it's worth thinking about what permutations are counting. Now, remember, we care, when we're talking about permutations, we care about who's sitting in which chair. So, for example, this is one permutation, and this would be counted as another permutation. And this would be counted as another permutation. This would be counted as another permutation. So notice, these are all the same three people, but we're putting them in different chairs. And this counted that. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
And this would be counted as another permutation. This would be counted as another permutation. So notice, these are all the same three people, but we're putting them in different chairs. And this counted that. That's counted in this 120. I could keep going. We could have that, or we could have that. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
And this counted that. That's counted in this 120. I could keep going. We could have that, or we could have that. So when we're thinking in the permutation world, we would count all of these, or we would count this as six different permutations. These are going towards this 120. And of course, we have other permutations where we involve other people, where we have, it could be FBC, FCB, FAC, F, F, actually, let me do it this way. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
We could have that, or we could have that. So when we're thinking in the permutation world, we would count all of these, or we would count this as six different permutations. These are going towards this 120. And of course, we have other permutations where we involve other people, where we have, it could be FBC, FCB, FAC, F, F, actually, let me do it this way. That would be a little bit more systematic. F, let me do it, BBFC, BCF, and obviously, I could keep going. I could do 120 of these. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
And of course, we have other permutations where we involve other people, where we have, it could be FBC, FCB, FAC, F, F, actually, let me do it this way. That would be a little bit more systematic. F, let me do it, BBFC, BCF, and obviously, I could keep going. I could do 120 of these. I'll do two more. You could have CFB, and then you could have CBF. So in the permutation world, these are literally 12 of the 120 permutations. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
I could do 120 of these. I'll do two more. You could have CFB, and then you could have CBF. So in the permutation world, these are literally 12 of the 120 permutations. But what if all we cared about is the three people we're choosing to sit down, but we don't care in what order that they're sitting or in which chair they're sitting? So in that world, these would all be one. This is all the same set of three people if we don't care which chair they're sitting in. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
So in the permutation world, these are literally 12 of the 120 permutations. But what if all we cared about is the three people we're choosing to sit down, but we don't care in what order that they're sitting or in which chair they're sitting? So in that world, these would all be one. This is all the same set of three people if we don't care which chair they're sitting in. This would also be the same set of three people. And so this question, if I have six people sitting in three chairs, how many ways can I choose three people out of the six where I don't care which chair they sit on? And I encourage you to pause the video and try to think of what that number would actually be. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
This is all the same set of three people if we don't care which chair they're sitting in. This would also be the same set of three people. And so this question, if I have six people sitting in three chairs, how many ways can I choose three people out of the six where I don't care which chair they sit on? And I encourage you to pause the video and try to think of what that number would actually be. Well, a big clue was, when we essentially wrote all of the permutations where we've picked a group of three people, we see that there's six ways of arranging the three people. And you pick a certain group of three people that turned into six permutations. And so if all you want to do is care about, well, how many different ways are there to choose three from the six, you would take your whole permutations, you would take your number of permutations, and then you would divide it by the number of ways to arrange three people. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
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