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So probability of 1, that's 3 eighths. That's right over there. That's 3 eighths. So let me draw that bar. Just like that. The probability that X equals 2 is also 3 eighths. So that's going to be that same level, just like that. | Constructing a probability distribution for random variable Khan Academy.mp3 |
So let me draw that bar. Just like that. The probability that X equals 2 is also 3 eighths. So that's going to be that same level, just like that. And then the probability that X equals 3, well, that's 1 eighth. So it's going to be the same height as this thing right over here. So actually, I'm using the wrong color. | Constructing a probability distribution for random variable Khan Academy.mp3 |
So that's going to be that same level, just like that. And then the probability that X equals 3, well, that's 1 eighth. So it's going to be the same height as this thing right over here. So actually, I'm using the wrong color. So it's going to look like this. It's going to look like this. And actually, let me just writ... | Constructing a probability distribution for random variable Khan Academy.mp3 |
So actually, I'm using the wrong color. So it's going to look like this. It's going to look like this. And actually, let me just write this a little bit neater. I can move that 3. So cut and paste. Let me move that 3 a little bit closer in, just so it looks a little bit neater. | Constructing a probability distribution for random variable Khan Academy.mp3 |
And actually, let me just write this a little bit neater. I can move that 3. So cut and paste. Let me move that 3 a little bit closer in, just so it looks a little bit neater. And I can move that 2 in, actually, as well. So cut and paste. So I can move that 2. | Constructing a probability distribution for random variable Khan Academy.mp3 |
Let me move that 3 a little bit closer in, just so it looks a little bit neater. And I can move that 2 in, actually, as well. So cut and paste. So I can move that 2. And there you have it. We have made a probability distribution for the random variable X. And the random variable X can only take on these discrete values... | Constructing a probability distribution for random variable Khan Academy.mp3 |
So I can move that 2. And there you have it. We have made a probability distribution for the random variable X. And the random variable X can only take on these discrete values. It can't take on the value half or the value pi or anything like that. And so what we've just done here is we've just constructed a discrete p... | Constructing a probability distribution for random variable Khan Academy.mp3 |
And the random variable X can only take on these discrete values. It can't take on the value half or the value pi or anything like that. And so what we've just done here is we've just constructed a discrete probability distribution. Let me write that down. So this right over here is a discrete. The random variable only... | Constructing a probability distribution for random variable Khan Academy.mp3 |
Perhaps you want to do this because over time, you're trying to breed watermelons that have fewer seeds, and you should see whether you are actually making progress. And you don't want to cut open every watermelon in your watermelon farm or patch or whatever it might be called, because you want to sell most of them. Yo... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
So let's start doing that. So let's say that you take these little cubic inch chunks out of a random sample of your watermelons, and then you count the number of seeds in them. And you have eight samples like this. So in one of them, you found four seeds. In the next, you found three, five, seven, two, nine, 11, and se... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
So in one of them, you found four seeds. In the next, you found three, five, seven, two, nine, 11, and seven. So this is a sample just to make sure we're visualizing it right. If this is a population of all of the chunks, these little cubic, I guess if we view this as a cubic inch, the cubic inch chunks in my entire wa... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
If this is a population of all of the chunks, these little cubic, I guess if we view this as a cubic inch, the cubic inch chunks in my entire watermelon farm, I'm sampling a very small sample of them. So I'm sampling a very small sample. Maybe I could have had a million over here. A million chunks of watermelon could h... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
A million chunks of watermelon could have been produced from my farm, but I'm only sampling. So capital N would be 1 million, lowercase n is equal to 8. And once again, you might want to have more samples, but this will make our math easy. Now, let's think about what statistics we can measure. Well, the first one that ... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Now, let's think about what statistics we can measure. Well, the first one that we often do is a measure of central tendency, and that's the arithmetic mean. But here, we were trying to estimate the population mean by coming up with the sample mean. So what is the sample mean going to be? Well, all we have to do is add... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
So what is the sample mean going to be? Well, all we have to do is add up these points, add up these measurements, and then divide by the number of measurements we have. So let's get our calculator out for that. Actually, maybe I don't need my calculator. Let's see. So 4 plus 3 is 7. 7 plus 5 is 12. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Actually, maybe I don't need my calculator. Let's see. So 4 plus 3 is 7. 7 plus 5 is 12. 12 plus 7 is 19. 19 plus 2 is 21. Plus 9 is 30. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
7 plus 5 is 12. 12 plus 7 is 19. 19 plus 2 is 21. Plus 9 is 30. Plus 11 is 41. Plus 7 is 48. So I'm going to get 48 over 8 data points. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Plus 9 is 30. Plus 11 is 41. Plus 7 is 48. So I'm going to get 48 over 8 data points. So this worked out quite well. 48 divided by 8 is equal to 6. So our sample mean is 6. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
So I'm going to get 48 over 8 data points. So this worked out quite well. 48 divided by 8 is equal to 6. So our sample mean is 6. It's our estimate of what the population mean might be. But we also want to think about how much in our population, we want to estimate how much in our population, how much spread is there? ... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
So our sample mean is 6. It's our estimate of what the population mean might be. But we also want to think about how much in our population, we want to estimate how much in our population, how much spread is there? How much do our measurements vary from this mean? So there we say, well, we can try to estimate the popul... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
How much do our measurements vary from this mean? So there we say, well, we can try to estimate the population variance by calculating the sample variance. And we're going to calculate the unbiased sample variance. Hopefully we're fairly convinced at this point why we divide by n minus 1. So we're going to calculate th... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Hopefully we're fairly convinced at this point why we divide by n minus 1. So we're going to calculate the unbiased sample variance. And if we do that, what do we get? Well, it's just going to be, I'll do this in a different color, it's going to be 4 minus 6 squared plus 3 minus 6 squared plus 5 minus 6 squared plus 7 ... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Well, it's just going to be, I'll do this in a different color, it's going to be 4 minus 6 squared plus 3 minus 6 squared plus 5 minus 6 squared plus 7 minus 6 squared plus 2 minus 6 squared plus 9 minus 6 squared plus 11 minus 6 squared plus 7 minus 6 squared. All of that divided by, not by 8. Remember, we want the un... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
We're going to divide it by 8 minus 1. So we're going to divide by 7. And so this is going to be equal to, let me give myself a little bit more real estate, the unbiased sample variance, and I could even denote it by this to make it clear that we're dividing by lowercase n minus 1, is going to be equal to, let's see, 4... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
That squared is positive 4. So I did that one. 3 minus 6 is negative 3. That squared is going to be 9. 5 minus 6 squared is 1 squared, which is 1. 7 minus 6 is, once again, 1 squared, which is 1. 2 minus 6, negative 4 squared, negative 4 squared is 16. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
That squared is going to be 9. 5 minus 6 squared is 1 squared, which is 1. 7 minus 6 is, once again, 1 squared, which is 1. 2 minus 6, negative 4 squared, negative 4 squared is 16. 9 minus 6 squared, well, that's going to be 9. 11 minus 6 squared, that is 25. And then finally, 7 minus 6 squared, that's another 1. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
2 minus 6, negative 4 squared, negative 4 squared is 16. 9 minus 6 squared, well, that's going to be 9. 11 minus 6 squared, that is 25. And then finally, 7 minus 6 squared, that's another 1. And we're going to divide it by 7. Now let's see if we can add this up in our heads. 4 plus 9 is 13. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
And then finally, 7 minus 6 squared, that's another 1. And we're going to divide it by 7. Now let's see if we can add this up in our heads. 4 plus 9 is 13. Plus 1 is 14, 15, 31, 40, 65, 66. So this is going to be equal to 66 over 7. And we could either divide. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
4 plus 9 is 13. Plus 1 is 14, 15, 31, 40, 65, 66. So this is going to be equal to 66 over 7. And we could either divide. That's 9 and 3 sevenths. We could write that as 9 and 3 sevenths. Or if we want to write that as a decimal, I can just take 66. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
And we could either divide. That's 9 and 3 sevenths. We could write that as 9 and 3 sevenths. Or if we want to write that as a decimal, I can just take 66. 66 divided by 7 gives us 9 point, I'll just round it. So it's approximately 9.43. So this is approximately 9.43. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Or if we want to write that as a decimal, I can just take 66. 66 divided by 7 gives us 9 point, I'll just round it. So it's approximately 9.43. So this is approximately 9.43. Now that gave us our unbiased sample variance. Well, how could we calculate a sample standard deviation? We want to somehow get at an estimate of... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
So this is approximately 9.43. Now that gave us our unbiased sample variance. Well, how could we calculate a sample standard deviation? We want to somehow get at an estimate of what the population standard deviation might be. Well, the logic should, I guess, is reasonable to say, well, this is our unbiased sample varia... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
We want to somehow get at an estimate of what the population standard deviation might be. Well, the logic should, I guess, is reasonable to say, well, this is our unbiased sample variance. It's our best estimate of what the true population variance is. When we think about population parameters to get the population sta... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
When we think about population parameters to get the population standard deviation, we just take the square root of the population variance. So if we want to get an estimate of the sample standard deviation, why don't we just take the square root of the unbiased sample variance? So that's what we'll do. So we'll define... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
So we'll define it that way. We'll call the sample standard deviation, we're going to define it to be equal to the square root of the unbiased sample variance. So it's going to be the square root of this quantity. And we could take our calculator out. It's going to be the square root of what I just typed in. I could do... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
And we could take our calculator out. It's going to be the square root of what I just typed in. I could do second answer. It'll be the last entry here. So the square root of that is, and I'll just round, it's approximately equal to 3.07. Now I'm going to tell you something very counterintuitive. Or at least initially i... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
It'll be the last entry here. So the square root of that is, and I'll just round, it's approximately equal to 3.07. Now I'm going to tell you something very counterintuitive. Or at least initially it's counterintuitive, but hopefully you'll appreciate this over time. This we've already talked about in some depth. Peopl... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Or at least initially it's counterintuitive, but hopefully you'll appreciate this over time. This we've already talked about in some depth. People have even created simulations to show that this is an unbiased estimate of population variance when we divide it by n minus 1. And that's a good starting point if we're goin... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
And that's a good starting point if we're going to take the square root of anything. But it actually turns out that because the square root function is nonlinear, that this sample standard deviation, and this is how it tends to be defined, sample standard deviation, that this sample standard deviation, which is the squ... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Because the square root function is nonlinear, it turns out that this is not an unbiased estimate of the true population standard deviation. And I encourage people to make simulations of that if they're interested. But then you might say, OK, well, we went through great pains to divide by n minus 1 here in order to get... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Why don't we go through similar pains and somehow figure out a formula for an unbiased estimate of the population standard deviation? And the reason why that's difficult is to unbiased the sample variance, we just have to divide by n minus 1 instead of n. And that worked for any probability distribution for our populat... | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Let's do a little bit of probability with playing cards. And for the sake of this video, we're going to assume that our deck has no jokers in it. You could do the same problems with the joker. You'll just get slightly different numbers. So with that out of the way, let's first just think about how many cards we have in... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
You'll just get slightly different numbers. So with that out of the way, let's first just think about how many cards we have in a standard playing deck. So you have four suits. So you have four suits. And the suits are the spades, the diamonds, the clubs, and the hearts. You have four suits. And then in each of those s... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So you have four suits. And the suits are the spades, the diamonds, the clubs, and the hearts. You have four suits. And then in each of those suits, you have 13 different types of cards. Or sometimes it's called the rank. So each suit has 13 types of cards. You have the ace, then you have the two, the three, the four, ... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
And then in each of those suits, you have 13 different types of cards. Or sometimes it's called the rank. So each suit has 13 types of cards. You have the ace, then you have the two, the three, the four, the five, the six, seven, eight, nine, 10. And then you have the jack, the king, and the queen. And that is 13 cards... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
You have the ace, then you have the two, the three, the four, the five, the six, seven, eight, nine, 10. And then you have the jack, the king, and the queen. And that is 13 cards. So for each suit, you can have any of these. For any of these, you can have any of the suits. So you could have a jack of diamonds, a jack o... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So for each suit, you can have any of these. For any of these, you can have any of the suits. So you could have a jack of diamonds, a jack of clubs, a jack of spades, or a jack of hearts. So if you just multiply these two things, you could take a deck of playing cards and actually count them, take out the jokers and co... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So if you just multiply these two things, you could take a deck of playing cards and actually count them, take out the jokers and count them. But if you just multiply this, you have four suits. Each of those suits have 13 types. So you're going to have 4 times 13 cards. Or you're going to have 52 cards in a standard pl... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So you're going to have 4 times 13 cards. Or you're going to have 52 cards in a standard playing deck. Another way you could say it, you're like, look, I'm going to have these ranks or types. And each of those come in four different suits, 13 times 4. Once again, you would have gotten 52 cards. Now with that out of the... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
And each of those come in four different suits, 13 times 4. Once again, you would have gotten 52 cards. Now with that out of the way, let's think about the probabilities of different events. So let's say I shuffle that deck. I shuffle it really, really well. And then I randomly pick a card from that deck. And I want to... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So let's say I shuffle that deck. I shuffle it really, really well. And then I randomly pick a card from that deck. And I want to think about what is the probability that I pick a jack. Well, how many equally likely events are there? Well, I could pick any one of those 52 cards. So there's 52 possibilities for when I p... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
And I want to think about what is the probability that I pick a jack. Well, how many equally likely events are there? Well, I could pick any one of those 52 cards. So there's 52 possibilities for when I pick that card. And how many of those 52 possibilities are jacks? Well, you have the jack of spades, the jack of diam... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So there's 52 possibilities for when I pick that card. And how many of those 52 possibilities are jacks? Well, you have the jack of spades, the jack of diamonds, the jack of clubs, and the jack of hearts. There's four jacks in that deck. So it is 4 over 52. These are both divisible by 4. 4 divided by 4 is 1. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
There's four jacks in that deck. So it is 4 over 52. These are both divisible by 4. 4 divided by 4 is 1. 52 divided by 4 is 13. Now let's think about the probability. So we're going to start over. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
4 divided by 4 is 1. 52 divided by 4 is 13. Now let's think about the probability. So we're going to start over. I'm going to put that jack back in. I'm going to reshuffle the deck. So once again, I still have 52 cards. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So we're going to start over. I'm going to put that jack back in. I'm going to reshuffle the deck. So once again, I still have 52 cards. So what's the probability that I get a hearts? What's the probability that I just randomly pick a card from a shuffled deck and it is a hearts? Its suit is a heart. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So once again, I still have 52 cards. So what's the probability that I get a hearts? What's the probability that I just randomly pick a card from a shuffled deck and it is a hearts? Its suit is a heart. Well, once again, there's 52 possible cards I could pick from, 52 possible equally likely events that we're dealing w... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Its suit is a heart. Well, once again, there's 52 possible cards I could pick from, 52 possible equally likely events that we're dealing with. And how many of those have our hearts? Well, essentially 13 of them are hearts. For each of those suits, you have 13 types. So there are 13 hearts in that deck. There are 13 dia... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Well, essentially 13 of them are hearts. For each of those suits, you have 13 types. So there are 13 hearts in that deck. There are 13 diamonds in that deck. There are 13 spades in that deck. There are 13 clubs in that deck. So 13 of the 52 would result in hearts. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
There are 13 diamonds in that deck. There are 13 spades in that deck. There are 13 clubs in that deck. So 13 of the 52 would result in hearts. And both of these are divisible by 13. This is the same thing as 1 fourth. 1 and 4 times, I will pick it out or I have a 1 and 4 probability of getting a hearts when I go to tha... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So 13 of the 52 would result in hearts. And both of these are divisible by 13. This is the same thing as 1 fourth. 1 and 4 times, I will pick it out or I have a 1 and 4 probability of getting a hearts when I go to that, when I randomly pick a card from that shuffle deck. Now let's do something that's a little bit more ... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
1 and 4 times, I will pick it out or I have a 1 and 4 probability of getting a hearts when I go to that, when I randomly pick a card from that shuffle deck. Now let's do something that's a little bit more interesting. Or maybe it's a little obvious. What's the probability that I pick something that is a jack? I'll just... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
What's the probability that I pick something that is a jack? I'll just write J. It's a jack and it is a hearts. Well, if you're reasonably familiar with cards, you'll know that there's actually only one card that is both a jack and a heart. It is literally the jack of hearts. So we're saying, what is the probability th... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Well, if you're reasonably familiar with cards, you'll know that there's actually only one card that is both a jack and a heart. It is literally the jack of hearts. So we're saying, what is the probability that we pick the exact card, the jack of hearts? Well, there's only one event, one card, that meets this criteria ... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Well, there's only one event, one card, that meets this criteria right over here. And there's 52 possible cards. So there's a 1 in 52 chance that I pick the jack of hearts, something that is both a jack and it's a heart. Now let's do something a little bit more interesting. What is the probability? You might want to pa... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Now let's do something a little bit more interesting. What is the probability? You might want to pause this and think about this a little bit before I give you the answer. What is the probability of, so I once again, I have a deck of 52 cards. I shuffle it, randomly pick a card from that deck. What is the probability t... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
What is the probability of, so I once again, I have a deck of 52 cards. I shuffle it, randomly pick a card from that deck. What is the probability that that card that I pick from that deck is a jack or a heart? So it could be the jack of hearts or it could be the jack of diamonds or it could be the jack of spades or it... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So it could be the jack of hearts or it could be the jack of diamonds or it could be the jack of spades or it could be the queen of hearts or it could be the two of hearts. So what is the probability of this? And this is a little bit more of an interesting thing because we know, first of all, that there are 52 possibil... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
There are 52 possibilities. But how many of those possibilities meet the criteria, meet these conditions that it is a jack or a heart? And to understand that, I'll draw a Venn diagram. Sounds kind of fancy, but nothing fancy here. So imagine that this rectangle I'm drawing here represents all of the outcomes. So if you... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Sounds kind of fancy, but nothing fancy here. So imagine that this rectangle I'm drawing here represents all of the outcomes. So if you want, you can imagine it has an area of 52. So this is 52 possible outcomes. Now, how many of those outcomes result in a jack? So we already learned, it's one out of 13 of those outcom... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So this is 52 possible outcomes. Now, how many of those outcomes result in a jack? So we already learned, it's one out of 13 of those outcomes result in a jack. So I could draw a little circle here where that area, and I'm approximating, that represents the probability of a jack. So it should be roughly 1 13th or 4 52n... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So I could draw a little circle here where that area, and I'm approximating, that represents the probability of a jack. So it should be roughly 1 13th or 4 52nds of this area right over here. So I'll just draw it like this. So this right over here is the probability of a jack. The probability of the jack. It is four, t... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So this right over here is the probability of a jack. The probability of the jack. It is four, there's four possible cards out of the 52. So that is four 52nds or one out of 13. 1 13. Now, what's the probability of getting a hearts? Well, I'll draw another little circle here that represents that. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So that is four 52nds or one out of 13. 1 13. Now, what's the probability of getting a hearts? Well, I'll draw another little circle here that represents that. 13 out of 52. 13 out of these 52 cards represent a heart. And actually, one of them represents both a heart and a jack. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Well, I'll draw another little circle here that represents that. 13 out of 52. 13 out of these 52 cards represent a heart. And actually, one of them represents both a heart and a jack. So I'm actually going to overlap them, and hopefully this will make sense in a second. So there's actually 13 cards that are a heart. S... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
And actually, one of them represents both a heart and a jack. So I'm actually going to overlap them, and hopefully this will make sense in a second. So there's actually 13 cards that are a heart. So this is the number of hearts. Number of hearts. And actually, let me write this top thing that way as well. That makes it... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So this is the number of hearts. Number of hearts. And actually, let me write this top thing that way as well. That makes it a little bit clearer that we're actually looking at. So let me clear that. So the number of jacks. Number of jacks. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
That makes it a little bit clearer that we're actually looking at. So let me clear that. So the number of jacks. Number of jacks. And of course, this overlap right here is the number of jacks and hearts. The number of items out of this 52 that are both a jack and a heart. It is in both sets here. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Number of jacks. And of course, this overlap right here is the number of jacks and hearts. The number of items out of this 52 that are both a jack and a heart. It is in both sets here. It is in this green circle, and it is in this orange circle. So this right over here, let me do that in yellow since I did that problem... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
It is in both sets here. It is in this green circle, and it is in this orange circle. So this right over here, let me do that in yellow since I did that problem in yellow. This right over here is the number of jacks and hearts. So let me draw a little arrow there. It's getting a little cluttered. Maybe I should have dr... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
This right over here is the number of jacks and hearts. So let me draw a little arrow there. It's getting a little cluttered. Maybe I should have drawn a little bit bigger. Number of jacks and hearts. Number of jacks and hearts. And that's an overlap over there. | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Maybe I should have drawn a little bit bigger. Number of jacks and hearts. Number of jacks and hearts. And that's an overlap over there. So what is the probability of getting a jack or a heart? So if you think about it, the probability is going to be the number of events that meet these conditions over the total number... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
And that's an overlap over there. So what is the probability of getting a jack or a heart? So if you think about it, the probability is going to be the number of events that meet these conditions over the total number of events. We already know the total number of events are 52, but how many meet these conditions? So i... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
We already know the total number of events are 52, but how many meet these conditions? So it's going to be the number, it's going to be, you could say, well, look, the green circle right there says the number that gives us a jack, and the orange circle tells us that the number that gives us a heart. So you might want t... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
But if you did that, you would be double counting. Because if you added up, if you just did four, if you did four plus 13, what are we saying? We're saying that there are four jacks, and we're saying that there are 13 hearts. But in both of these, when we do it this way, in both cases, we are counting the jack of heart... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
But in both of these, when we do it this way, in both cases, we are counting the jack of hearts. We're putting the jack of hearts here, and we're putting the jack of hearts here. So we're counting the jack of hearts twice, even though there's only one card there. So you would have to subtract out where they're common. ... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So you would have to subtract out where they're common. You would have to subtract out the item that is both a jack and a heart. So you would subtract out a one. Another way to think about it is, you really want to figure out the total area here. You want to figure out the total area here. You want to figure out this t... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
Another way to think about it is, you really want to figure out the total area here. You want to figure out the total area here. You want to figure out this total area. And let me zoom in, and I'll generalize it a little bit. So if you have one circle like that, and then you have another overlapping circle like that, a... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
And let me zoom in, and I'll generalize it a little bit. So if you have one circle like that, and then you have another overlapping circle like that, and you wanted to figure out the total area of the circles combined, you would look at the area of this circle, you would look at the area of this circle, and then you co... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
So in order to only count that area once, you have to subtract that area from the sum. So if this area has A, this area is B, and the intersection where they overlap is C, the combined area is going to be A plus B minus where they overlap, minus C. So that's the same thing over here. We're counting all the jacks, and t... | Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3 |
He randomly selects 30 of the 150 total seniors and finds that seven of those sampled would order the vegetarian option. Which conditions for constructing this confidence interval did Ali's sample meet? So pause this video, and you can select more than one of these. All right, now let's work through this together. So o... | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
All right, now let's work through this together. So one thing that you might be wondering is, well, what is a one-sample z-interval? Well, you could really interpret that as he's gonna take one sample and then construct a confidence interval based on that. The reason why it might be called a z-interval is the whole ide... | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
The reason why it might be called a z-interval is the whole idea behind a confidence interval is you're going to pick a number of standard deviations above and below the true parameter that you are actually trying to estimate, and then use that to make your inferences. And one way of thinking about the number of standa... | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
But remember, in order to construct a confidence interval, we have to make some assumptions. He's taking, there's 150 students right over here. He's finding it impractical to survey all 150 to figure out the true population proportion. So instead, he samples 30 of the seniors. So n is equal to 30. And from that, he cal... | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
So instead, he samples 30 of the seniors. So n is equal to 30. And from that, he calculates a sample proportion. It looks like seven out of the 30 are they want the vegetarian option. And he's going to determine some confidence level and then construct a confidence interval. But remember the conditions that we've talke... | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
It looks like seven out of the 30 are they want the vegetarian option. And he's going to determine some confidence level and then construct a confidence interval. But remember the conditions that we've talked about in previous videos. The first thing is we have to be confident that is this a random sample? So that woul... | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
The first thing is we have to be confident that is this a random sample? So that would be the random condition. And that's what choice A is telling us. The data is a random sample from the population of interest. Do we know that? Well, it tells us in the passage here, he randomly selects 30 of the total seniors. So I g... | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
The data is a random sample from the population of interest. Do we know that? Well, it tells us in the passage here, he randomly selects 30 of the total seniors. So I guess we'll take their word for it. We don't know his methodology of what he considers random, but we'll take their word for it that yes, this has been m... | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
So I guess we'll take their word for it. We don't know his methodology of what he considers random, but we'll take their word for it that yes, this has been met. The data is a random, random sample. If it said he sampled the football team, well, that would not have been a random sample. The next condition here, it look... | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
If it said he sampled the football team, well, that would not have been a random sample. The next condition here, it looks all mathematical, but this is really the normal condition. And the idea behind the normal condition is that in order to construct these confidence intervals, we're assuming that the sampling distri... | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
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