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We want to know the probability that our sample will have an average greater than 2.2. So this is the distribution of all of the possible samples, the means of all of the possible samples. Now to be greater than 2.2, 2.2 is going to be right around here. 2.2 is going to be right around here. So we are essentially asking, we will run out if our sample mean falls into this bucket over here. So we essentially need to figure out what is, you can even view it as, what's this area under this curve there. And to figure that out, we just have to figure out how many standard deviations above the mean we are, which is going to be our z-score. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
2.2 is going to be right around here. So we are essentially asking, we will run out if our sample mean falls into this bucket over here. So we essentially need to figure out what is, you can even view it as, what's this area under this curve there. And to figure that out, we just have to figure out how many standard deviations above the mean we are, which is going to be our z-score. And then we could use a z-table to figure out what this area right over here is. So we want to know, when we are above 2.2 liters, so 2.2 liters, we could even do it in our head, 2.2 liters is what we care about. That's right over here. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
And to figure that out, we just have to figure out how many standard deviations above the mean we are, which is going to be our z-score. And then we could use a z-table to figure out what this area right over here is. So we want to know, when we are above 2.2 liters, so 2.2 liters, we could even do it in our head, 2.2 liters is what we care about. That's right over here. That is, our mean is 2, so we are 0.2 above the mean. And if we want that in terms of standard deviations, we just divide this by the standard deviation of this distribution over here. And we figured out what that is. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
That's right over here. That is, our mean is 2, so we are 0.2 above the mean. And if we want that in terms of standard deviations, we just divide this by the standard deviation of this distribution over here. And we figured out what that is. The standard deviation of this distribution is 0.099. So if we take, and you'll see a formula where you take this value minus the mean and divide it by the standard deviation, that's all we're doing. We're just figuring out how many standard deviations above the mean we are. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
And we figured out what that is. The standard deviation of this distribution is 0.099. So if we take, and you'll see a formula where you take this value minus the mean and divide it by the standard deviation, that's all we're doing. We're just figuring out how many standard deviations above the mean we are. So you just take this number right over here, divided by the standard deviation, so 0.099, or 0.099. And then we get, let's get our calculator. And actually, we had the exact number over here. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
We're just figuring out how many standard deviations above the mean we are. So you just take this number right over here, divided by the standard deviation, so 0.099, or 0.099. And then we get, let's get our calculator. And actually, we had the exact number over here. So we could just take 0.2, we could just take this 0.2, 0.2, divided by this value over here. And on this calculator, when I press second answer, it just means the last answer. So I'm taking 0.2 divided by this value over there. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
And actually, we had the exact number over here. So we could just take 0.2, we could just take this 0.2, 0.2, divided by this value over here. And on this calculator, when I press second answer, it just means the last answer. So I'm taking 0.2 divided by this value over there. And I get 2.020. So this means, so that means that this value, or I should write this probability is the same probability of being 2.02 standard deviations. Maybe I should write it this way. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So I'm taking 0.2 divided by this value over there. And I get 2.020. So this means, so that means that this value, or I should write this probability is the same probability of being 2.02 standard deviations. Maybe I should write it this way. More than, let me write it down here where I have more space. So this all boils down to the probability of running out of water is the probability that the sample mean will be more than just the 50 that we happen to select. Remember, if we take a bunch of samples of 50 and plot all of them, we'll get this whole distribution. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
Maybe I should write it this way. More than, let me write it down here where I have more space. So this all boils down to the probability of running out of water is the probability that the sample mean will be more than just the 50 that we happen to select. Remember, if we take a bunch of samples of 50 and plot all of them, we'll get this whole distribution. But the 150, the group of 50 that we happen to select, the probability of running out of water is the same thing as the probability of the mean of those people will be more than 2.020 standard deviations above the mean. Above the mean of this distribution, which is actually the same distribution. So what is that going to be? | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
Remember, if we take a bunch of samples of 50 and plot all of them, we'll get this whole distribution. But the 150, the group of 50 that we happen to select, the probability of running out of water is the same thing as the probability of the mean of those people will be more than 2.020 standard deviations above the mean. Above the mean of this distribution, which is actually the same distribution. So what is that going to be? And here we just have to look up our z table. Remember, this 2.02 is just this value right here. 0.2 divided by 0.09. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So what is that going to be? And here we just have to look up our z table. Remember, this 2.02 is just this value right here. 0.2 divided by 0.09. I just had to pause the video because there's some type of fighter jet outside or something. Anyway, hopefully they won't come back. But anyway, so we need to figure out the probability that the sample mean will be more than 2.02 standard deviations above the mean. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
0.2 divided by 0.09. I just had to pause the video because there's some type of fighter jet outside or something. Anyway, hopefully they won't come back. But anyway, so we need to figure out the probability that the sample mean will be more than 2.02 standard deviations above the mean. And to figure that out, we go to a z table. And you can find this pretty much anywhere. Usually it's in any stat book or on the internet, wherever. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
But anyway, so we need to figure out the probability that the sample mean will be more than 2.02 standard deviations above the mean. And to figure that out, we go to a z table. And you can find this pretty much anywhere. Usually it's in any stat book or on the internet, wherever. And so essentially, we want to know the probability. The z table will tell you how much area is below this value. So if you go to z of 2.02, that was the value that we were dealing with, right? | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
Usually it's in any stat book or on the internet, wherever. And so essentially, we want to know the probability. The z table will tell you how much area is below this value. So if you go to z of 2.02, that was the value that we were dealing with, right? Yeah, 2.02. It was, so you go for the first digit. We go to 2.0, and it was 2.02. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So if you go to z of 2.02, that was the value that we were dealing with, right? Yeah, 2.02. It was, so you go for the first digit. We go to 2.0, and it was 2.02. 2.02 is right over there, right? So we had 2.0, and then the next digit you go up here. So it's 2.02 is right over there. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
We go to 2.0, and it was 2.02. 2.02 is right over there, right? So we had 2.0, and then the next digit you go up here. So it's 2.02 is right over there. So this 0.9783, let me write it down over here. This 0.9783, I want to be very careful, 0.9783. That z table, that's not this value over here. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So it's 2.02 is right over there. So this 0.9783, let me write it down over here. This 0.9783, I want to be very careful, 0.9783. That z table, that's not this value over here. This 0.9783 on the z table, that is giving us this whole area over here. It's giving us the probability that we are below that value, that we are less than 2.02 standard deviations above the mean. So it's giving us that value over here. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
That z table, that's not this value over here. This 0.9783 on the z table, that is giving us this whole area over here. It's giving us the probability that we are below that value, that we are less than 2.02 standard deviations above the mean. So it's giving us that value over here. So to answer our question, to answer this probability, we just have to subtract this from 1, because these will all add up to 1. So we just take our calculator back out, and we just take 1 minus 0.9783 is equal to 0.0217. So this right here is 0.0217. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So it's giving us that value over here. So to answer our question, to answer this probability, we just have to subtract this from 1, because these will all add up to 1. So we just take our calculator back out, and we just take 1 minus 0.9783 is equal to 0.0217. So this right here is 0.0217. Or another way you could say it, it is a 2.17% probability that we will run out of water. And we are done. Let me make sure I got that number right. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
Let's say that you have a cholesterol test, and you know, you somehow magically know that the probability that it is accurate, that it gives the correct results, is 99, 99%. You have a 99 out of 100 chance that any time you apply this test, that it is going to be accurate. Now let's say that you, and you just magically know that, we're just assuming that. Now let's just say that you get 100 folks into this room, and you apply this test to all 100 of them. So apply, apply test 100 times. So what are some of the possible outcomes here? Is it for sure that 99, exactly 99 out of 100 are going to be accurate, and that one out of 100 is going to be inaccurate? | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
Now let's just say that you get 100 folks into this room, and you apply this test to all 100 of them. So apply, apply test 100 times. So what are some of the possible outcomes here? Is it for sure that 99, exactly 99 out of 100 are going to be accurate, and that one out of 100 is going to be inaccurate? Well that's definitely a likely possibility, but it's also possible you get a little lucky and all 100 are accurate, or you get a little unlucky, and that 98 are accurate, and that two are inaccurate. And actually I calculated the probabilities ahead of time, and the goal of this video isn't to go into the probability and combinatorics of it, but if you're curious about it, there's a lot of good videos on probability and combinatorics on Khan Academy. But I calculated it ahead of time, and the probability, if you have something that has a 99% chance of being accurate, and you apply it 100 times, the probability that it is accurate, that it is accurate 100 out of the 100 times, is approximately equal to, approximately equal to 36.6%. | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
Is it for sure that 99, exactly 99 out of 100 are going to be accurate, and that one out of 100 is going to be inaccurate? Well that's definitely a likely possibility, but it's also possible you get a little lucky and all 100 are accurate, or you get a little unlucky, and that 98 are accurate, and that two are inaccurate. And actually I calculated the probabilities ahead of time, and the goal of this video isn't to go into the probability and combinatorics of it, but if you're curious about it, there's a lot of good videos on probability and combinatorics on Khan Academy. But I calculated it ahead of time, and the probability, if you have something that has a 99% chance of being accurate, and you apply it 100 times, the probability that it is accurate, that it is accurate 100 out of the 100 times, is approximately equal to, approximately equal to 36.6%. I rounded to the nearest tenth of a percent. So it's a little better than a third chance that you'll actually get all of, all of the people are going to get an accurate result, even though for any one of them, there's a 99% chance that it's accurate. Now we could keep going. | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
But I calculated it ahead of time, and the probability, if you have something that has a 99% chance of being accurate, and you apply it 100 times, the probability that it is accurate, that it is accurate 100 out of the 100 times, is approximately equal to, approximately equal to 36.6%. I rounded to the nearest tenth of a percent. So it's a little better than a third chance that you'll actually get all of, all of the people are going to get an accurate result, even though for any one of them, there's a 99% chance that it's accurate. Now we could keep going. The probability that it is accurate, I'm just going to put these quotes here so I don't have to rewrite accurate over and over again. The probability that it is accurate 99 out of 100 times, I calculated it ahead of time, it is approximately 37.0%. So this is what you would expect. | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
Now we could keep going. The probability that it is accurate, I'm just going to put these quotes here so I don't have to rewrite accurate over and over again. The probability that it is accurate 99 out of 100 times, I calculated it ahead of time, it is approximately 37.0%. So this is what you would expect. Getting 100 out of 100 doesn't seem that unlikely if each of the times you apply it has a 99% chance of being accurate. But it makes sense that you, that you would expect 99 out of 100 to be even more likely, slightly more likely. And we can of course keep going. | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
So this is what you would expect. Getting 100 out of 100 doesn't seem that unlikely if each of the times you apply it has a 99% chance of being accurate. But it makes sense that you, that you would expect 99 out of 100 to be even more likely, slightly more likely. And we can of course keep going. The probability that it is accurate 98 out of 100 times is approximately 18.5%. And I'm just going to do a few more. The probability that it is accurate 97 out of 100 times, and once again I calculated all of these ahead of time, is 6%. | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
And we can of course keep going. The probability that it is accurate 98 out of 100 times is approximately 18.5%. And I'm just going to do a few more. The probability that it is accurate 97 out of 100 times, and once again I calculated all of these ahead of time, is 6%. So it's definitely in the realm of possibility, but it's, the probability is much lower than getting, having 99 out of 100 or 100 out of 100 being accurate. And then the probability, let me put the double quotes here, the probability that it's accurate 96 out of 100 times is approximately 1.5%. And then the probability, and I'll just do one more, and I could keep going, the probability, there's some probability that even though each test has a 99% chance, you just get super unlucky and that very few of them are accurate. | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
The probability that it is accurate 97 out of 100 times, and once again I calculated all of these ahead of time, is 6%. So it's definitely in the realm of possibility, but it's, the probability is much lower than getting, having 99 out of 100 or 100 out of 100 being accurate. And then the probability, let me put the double quotes here, the probability that it's accurate 96 out of 100 times is approximately 1.5%. And then the probability, and I'll just do one more, and I could keep going, the probability, there's some probability that even though each test has a 99% chance, you just get super unlucky and that very few of them are accurate. But I'll just, and you see, you see what's happening to the probabilities as we have fewer and fewer of them being accurate. It becomes less and less probable. So the probability that 95 out of the 100 are accurate is approximately 0.3%. | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
And then the probability, and I'll just do one more, and I could keep going, the probability, there's some probability that even though each test has a 99% chance, you just get super unlucky and that very few of them are accurate. But I'll just, and you see, you see what's happening to the probabilities as we have fewer and fewer of them being accurate. It becomes less and less probable. So the probability that 95 out of the 100 are accurate is approximately 0.3%. So this was just kind of a, I guess you could say a thought experiment. If we had a test that we know for sure that every time you administer it the probability that it's accurate is 99%, then these are the probabilities that if you administer it 100 times that you get 100 out of 100 accurate, the probability that you get 99 out of 100 accurate, and so on and so forth. So let's just keep that in mind and then think a little bit about hypothesis testing and how we can use this framework. | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
So the probability that 95 out of the 100 are accurate is approximately 0.3%. So this was just kind of a, I guess you could say a thought experiment. If we had a test that we know for sure that every time you administer it the probability that it's accurate is 99%, then these are the probabilities that if you administer it 100 times that you get 100 out of 100 accurate, the probability that you get 99 out of 100 accurate, and so on and so forth. So let's just keep that in mind and then think a little bit about hypothesis testing and how we can use this framework. So let's put all that in the back of our minds. And let's say that you have devised a new test. You have a new test and you don't know how accurate it is. | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
So let's just keep that in mind and then think a little bit about hypothesis testing and how we can use this framework. So let's put all that in the back of our minds. And let's say that you have devised a new test. You have a new test and you don't know how accurate it is. You have a new cholesterol test. You don't know how accurate it is. You know that in order for it to be approved by whatever governing body, it has to be accurate 99, the probability of it being accurate has to be 99%. | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
You have a new test and you don't know how accurate it is. You have a new cholesterol test. You don't know how accurate it is. You know that in order for it to be approved by whatever governing body, it has to be accurate 99, the probability of it being accurate has to be 99%. So needs to have probability of accurate equal to 99%. You don't know if this is true. You just know that that's what it needs to be. | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
You know that in order for it to be approved by whatever governing body, it has to be accurate 99, the probability of it being accurate has to be 99%. So needs to have probability of accurate equal to 99%. You don't know if this is true. You just know that that's what it needs to be. And so you have your test and let's say you set up a hypothesis and your hypothesis could be a lot of things and once you get deeper into statistics, there's null hypothesis and alternate hypotheses, but let's just start with just a simple hypothesis. You're hopeful your hypothesis is that the probability that your new test is accurate is this is your hypothesis because you want that to be your hypothesis because if you feel good about it, then you're like, okay, maybe I'll get approved by the appropriate governing body. So you say, hey, my hypothesis is that my new test is accurate 99, the probability of it being accurate is 99%. | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
You just know that that's what it needs to be. And so you have your test and let's say you set up a hypothesis and your hypothesis could be a lot of things and once you get deeper into statistics, there's null hypothesis and alternate hypotheses, but let's just start with just a simple hypothesis. You're hopeful your hypothesis is that the probability that your new test is accurate is this is your hypothesis because you want that to be your hypothesis because if you feel good about it, then you're like, okay, maybe I'll get approved by the appropriate governing body. So you say, hey, my hypothesis is that my new test is accurate 99, the probability of it being accurate is 99%. So then you go off and you apply it 100 times. So you apply your new test. You don't know the actual probability of it being accurate. | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
So you say, hey, my hypothesis is that my new test is accurate 99, the probability of it being accurate is 99%. So then you go off and you apply it 100 times. So you apply your new test. You don't know the actual probability of it being accurate. You apply the test 100 times. And let's say out of those 100 times, you get that they are accurate, you get that it is accurate and you're able to use some other test that you know, some for sure test, some super accurate test to verify your own test results and you see that it is accurate 95 out of the 100 times. So the question you have is, well, does the hypothesis make sense to you? | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
You don't know the actual probability of it being accurate. You apply the test 100 times. And let's say out of those 100 times, you get that they are accurate, you get that it is accurate and you're able to use some other test that you know, some for sure test, some super accurate test to verify your own test results and you see that it is accurate 95 out of the 100 times. So the question you have is, well, does the hypothesis make sense to you? Will you accept this hypothesis? Well, what you say is, well, if my hypothesis was true, if my test were accurate 99, if the probability of my test being accurate is 99%, what's the probability of me getting this outcome? Well, we figured that out. | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
So the question you have is, well, does the hypothesis make sense to you? Will you accept this hypothesis? Well, what you say is, well, if my hypothesis was true, if my test were accurate 99, if the probability of my test being accurate is 99%, what's the probability of me getting this outcome? Well, we figured that out. If it really was accurate 99% of the time, then the probability of getting this outcome is only 0.3%. So if you assume true, if you assume hypothesis, I'll just write hype. If you assume the hypothesis is true, the probability of the outcome you got, probability of observed outcome is approximately 0.3%. | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
Well, we figured that out. If it really was accurate 99% of the time, then the probability of getting this outcome is only 0.3%. So if you assume true, if you assume hypothesis, I'll just write hype. If you assume the hypothesis is true, the probability of the outcome you got, probability of observed outcome is approximately 0.3%. And so you say, look, you know, maybe it's definitely possible that I just got very, very, very, very unlikely, but based on this, I probably should reject my hypothesis because the probability of me getting this outcome, if the hypothesis was true, is very, very, very, very low. And as we go deeper into statistics, you'll see that there are thresholds that people often set for, you know, if the probability of something happening or not happening is above or below some threshold, then we might reject a certain hypothesis. But in this world, you could see that, look, if my test really was accurate 99% of the time, for me to get, when I apply it to 100 people, it's only accurate 95 out of 100. | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
If you assume the hypothesis is true, the probability of the outcome you got, probability of observed outcome is approximately 0.3%. And so you say, look, you know, maybe it's definitely possible that I just got very, very, very, very unlikely, but based on this, I probably should reject my hypothesis because the probability of me getting this outcome, if the hypothesis was true, is very, very, very, very low. And as we go deeper into statistics, you'll see that there are thresholds that people often set for, you know, if the probability of something happening or not happening is above or below some threshold, then we might reject a certain hypothesis. But in this world, you could see that, look, if my test really was accurate 99% of the time, for me to get, when I apply it to 100 people, it's only accurate 95 out of 100. If my hypothesis is true, that would have only, there's only a 0.3% chance that I would have seen this observation. So based on that, it might be completely reasonable to say, you know what, I might reject my hypothesis. Look for a new test. | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
I actually recorded this video earlier today, but then I realized my microphone wasn't plugged in, and I won't name names in terms of who unplugged it. But anyway, back to probability. My wife is giggling mischievously. Anyway, so let's do a slightly harder problem than we did before. We were dealing with fair coins. Let's deal with a slightly unfair coin. Let's say I have a coin, and it's a, actually instead of unfair coin, let's do basketball. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
Anyway, so let's do a slightly harder problem than we did before. We were dealing with fair coins. Let's deal with a slightly unfair coin. Let's say I have a coin, and it's a, actually instead of unfair coin, let's do basketball. Let's say I'm shooting free throws, and I have a free throw percentage of 80%. So when I shoot a free throw 8 out of 10 times, or 80% of the time, I will make it. But that also says that 20% of the time, I will miss it. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
Let's say I have a coin, and it's a, actually instead of unfair coin, let's do basketball. Let's say I'm shooting free throws, and I have a free throw percentage of 80%. So when I shoot a free throw 8 out of 10 times, or 80% of the time, I will make it. But that also says that 20% of the time, I will miss it. So given that, if I were to take, I don't know, 5 free throws, what is the probability that I make at least 3 of the 5 free throws? Let's think of it this way. What is the probability of any particular combination of making 3 out of the 5? | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
But that also says that 20% of the time, I will miss it. So given that, if I were to take, I don't know, 5 free throws, what is the probability that I make at least 3 of the 5 free throws? Let's think of it this way. What is the probability of any particular combination of making 3 out of the 5? So what do I mean by that? Let me pick a particular combination. Let's say it's a basket, basket, basket, and then I miss, miss. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
What is the probability of any particular combination of making 3 out of the 5? So what do I mean by that? Let me pick a particular combination. Let's say it's a basket, basket, basket, and then I miss, miss. So that would be, I made 3 out of the 5. It could be, I don't know, basket, miss, basket, miss, basket, and there's a bunch of them, and we'll actually try to figure out how many of them there are. But what is the probability of this particular combination? | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
Let's say it's a basket, basket, basket, and then I miss, miss. So that would be, I made 3 out of the 5. It could be, I don't know, basket, miss, basket, miss, basket, and there's a bunch of them, and we'll actually try to figure out how many of them there are. But what is the probability of this particular combination? Well, I have an 80% chance of making this first basket times 80%. That's a times right there. Times 80%. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
But what is the probability of this particular combination? Well, I have an 80% chance of making this first basket times 80%. That's a times right there. Times 80%. And then what's my probability of missing? Well, that's 20%, right? Times 0.2, times 0.2. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
Times 80%. And then what's my probability of missing? Well, that's 20%, right? Times 0.2, times 0.2. So this equals 0.8 to the third power times 0.2 squared. Well, what's the probability of getting this exact combination? Well, it's 0.8 times, and then I miss. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
Times 0.2, times 0.2. So this equals 0.8 to the third power times 0.2 squared. Well, what's the probability of getting this exact combination? Well, it's 0.8 times, and then I miss. There's a 20% chance of that. So times 0.2, times 0.8, times 0.2, times 0.8, right? We could rearrange this, because when you multiply numbers, it doesn't matter what order you multiply them in. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
Well, it's 0.8 times, and then I miss. There's a 20% chance of that. So times 0.2, times 0.8, times 0.2, times 0.8, right? We could rearrange this, because when you multiply numbers, it doesn't matter what order you multiply them in. So this is the same thing as 0.8 times 0.8 times 0.8 times 0.2 times 0.2. So this is also the same thing as 0.8 to the third times 0.2 squared. So pretty much any particular, the probability of getting any particular combination of three baskets and two misses is going to be 0.8 to the third times 0.2 squared. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
We could rearrange this, because when you multiply numbers, it doesn't matter what order you multiply them in. So this is the same thing as 0.8 times 0.8 times 0.8 times 0.2 times 0.2. So this is also the same thing as 0.8 to the third times 0.2 squared. So pretty much any particular, the probability of getting any particular combination of three baskets and two misses is going to be 0.8 to the third times 0.2 squared. Now, what's the total probability of getting 3 out of 5? Well, it's going to be the sum of all of these combinations. You know, I could list them all, but we, hopefully now, are proficient enough in combinatorics and combinations to figure out how many different ways, if we have 5 baskets and we're picking, or we have 5 shots, and we're picking 3 of them to be the ones that are basket shots. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
So pretty much any particular, the probability of getting any particular combination of three baskets and two misses is going to be 0.8 to the third times 0.2 squared. Now, what's the total probability of getting 3 out of 5? Well, it's going to be the sum of all of these combinations. You know, I could list them all, but we, hopefully now, are proficient enough in combinatorics and combinations to figure out how many different ways, if we have 5 baskets and we're picking, or we have 5 shots, and we're picking 3 of them to be the ones that are basket shots. What do I mean? So let's say my 5 shots, so I have shot 1, 2, 3, 4, 5. And I'm going to, out of these 5, I'm going to choose 3. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
You know, I could list them all, but we, hopefully now, are proficient enough in combinatorics and combinations to figure out how many different ways, if we have 5 baskets and we're picking, or we have 5 shots, and we're picking 3 of them to be the ones that are basket shots. What do I mean? So let's say my 5 shots, so I have shot 1, 2, 3, 4, 5. And I'm going to, out of these 5, I'm going to choose 3. So I'm, once again, I'm putting my hat on as the god of probability. And I will choose 3 of these shots to be the ones that happen to be the ones that get made. So essentially, out of 5, I am choosing 3. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
And I'm going to, out of these 5, I'm going to choose 3. So I'm, once again, I'm putting my hat on as the god of probability. And I will choose 3 of these shots to be the ones that happen to be the ones that get made. So essentially, out of 5, I am choosing 3. 5 choose 3. And what does that equal to? That's 5 factorial over 3 factorial times 5 minus 3 factorial, so that's 2 factorial. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
So essentially, out of 5, I am choosing 3. 5 choose 3. And what does that equal to? That's 5 factorial over 3 factorial times 5 minus 3 factorial, so that's 2 factorial. That equals 5 times 4 times 3 times 2 times 1 over 3 times 2 times 1 times 2 times 1. We can ignore all the 1's. Let's see, we get 3 times 2 times 1, 3 times 2 times 1, we can cancel that. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
That's 5 factorial over 3 factorial times 5 minus 3 factorial, so that's 2 factorial. That equals 5 times 4 times 3 times 2 times 1 over 3 times 2 times 1 times 2 times 1. We can ignore all the 1's. Let's see, we get 3 times 2 times 1, 3 times 2 times 1, we can cancel that. This 1 we can ignore. And then this 2, and then this turns into 2. So there are 10 possible combinations. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
Let's see, we get 3 times 2 times 1, 3 times 2 times 1, we can cancel that. This 1 we can ignore. And then this 2, and then this turns into 2. So there are 10 possible combinations. These are 2 of them. Basket, basket, basket, miss, miss, basket, miss, basket, miss, basket. And it's a good exercise for you to list the other 8 of them. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
So there are 10 possible combinations. These are 2 of them. Basket, basket, basket, miss, miss, basket, miss, basket, miss, basket. And it's a good exercise for you to list the other 8 of them. But using just the binomial coefficient, and hopefully you have an intuition of why that works, and I'd be happy to make more videos if you feel that you need more explanation, but I made a couple. There are 10 combinations. So essentially, the probability of getting exactly 3 out of 5 baskets if I am an 80% free throw shot is going to be, let me switch colors, the probability of 3 out of 5 baskets is going to be equal to the probability of each of the combinations, which is 0.8 to the third times 0.2 squared, right? | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
And it's a good exercise for you to list the other 8 of them. But using just the binomial coefficient, and hopefully you have an intuition of why that works, and I'd be happy to make more videos if you feel that you need more explanation, but I made a couple. There are 10 combinations. So essentially, the probability of getting exactly 3 out of 5 baskets if I am an 80% free throw shot is going to be, let me switch colors, the probability of 3 out of 5 baskets is going to be equal to the probability of each of the combinations, which is 0.8 to the third times 0.2 squared, right? I make 3, miss 2, and then times the total number of combinations, right? Each of these has a probability of this much. And then there's 10 different arrangements that I could make. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
So essentially, the probability of getting exactly 3 out of 5 baskets if I am an 80% free throw shot is going to be, let me switch colors, the probability of 3 out of 5 baskets is going to be equal to the probability of each of the combinations, which is 0.8 to the third times 0.2 squared, right? I make 3, miss 2, and then times the total number of combinations, right? Each of these has a probability of this much. And then there's 10 different arrangements that I could make. There's 10 different ways of getting 3 baskets and 2 misses. So times 10. And what is that equal to? | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
And then there's 10 different arrangements that I could make. There's 10 different ways of getting 3 baskets and 2 misses. So times 10. And what is that equal to? Let me get my high-end calculator here. So let's see what that is. That is 0.8 times 0.8 times 0.8 times 0.2 times 0.2 times 10 equals 20.48. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
And what is that equal to? Let me get my high-end calculator here. So let's see what that is. That is 0.8 times 0.8 times 0.8 times 0.2 times 0.2 times 10 equals 20.48. So it's essentially a 20.48% chance that I get exactly 3 out of 5 of the baskets. Now let's make it slightly more interesting. Let's say I don't want to know the probability of 3 out of 5. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
That is 0.8 times 0.8 times 0.8 times 0.2 times 0.2 times 10 equals 20.48. So it's essentially a 20.48% chance that I get exactly 3 out of 5 of the baskets. Now let's make it slightly more interesting. Let's say I don't want to know the probability of 3 out of 5. And this is actually something that probably people are more likely to ask. What is the probability of getting at least 3 baskets? Well, if you think about it, this is the probability. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
Let's say I don't want to know the probability of 3 out of 5. And this is actually something that probably people are more likely to ask. What is the probability of getting at least 3 baskets? Well, if you think about it, this is the probability. This is equal to the probability of getting 3 out of 5 baskets plus the probability of getting exactly 4 out of 5 baskets plus the probability of getting exactly 5 out of 5 baskets, right? Well, we already figured this one out. That's 20.48%. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
Well, if you think about it, this is the probability. This is equal to the probability of getting 3 out of 5 baskets plus the probability of getting exactly 4 out of 5 baskets plus the probability of getting exactly 5 out of 5 baskets, right? Well, we already figured this one out. That's 20.48%. So what's the probability of getting 4 out of 5 baskets? Well, once again, if we want exactly 4 out of 5 baskets, so an example could be, I don't know, basket, basket, basket, basket, what's the probability of any one of the combinations where I make 4 baskets? Well, it's going to be 0.8 to the 4th times my, and then I have a 20% chance of that one miss, right? | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
That's 20.48%. So what's the probability of getting 4 out of 5 baskets? Well, once again, if we want exactly 4 out of 5 baskets, so an example could be, I don't know, basket, basket, basket, basket, what's the probability of any one of the combinations where I make 4 baskets? Well, it's going to be 0.8 to the 4th times my, and then I have a 20% chance of that one miss, right? And it could have been basket, miss, basket, basket, basket. Right? That's the same. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
Well, it's going to be 0.8 to the 4th times my, and then I have a 20% chance of that one miss, right? And it could have been basket, miss, basket, basket, basket. Right? That's the same. That's also exactly 4. But when you multiply them, the probability of getting any one of these particular combinations is exactly this, 0.8 to the 4th times 0.2. And so how many ways can you, if I have 5 baskets, how many ways can I pick 4 of them to be the ones that I make if I'm once again the god of probability? | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
That's the same. That's also exactly 4. But when you multiply them, the probability of getting any one of these particular combinations is exactly this, 0.8 to the 4th times 0.2. And so how many ways can you, if I have 5 baskets, how many ways can I pick 4 of them to be the ones that I make if I'm once again the god of probability? So this is going to be 0.8 to the 4th times 0.2 times out of 5 baskets, I'm choosing 4 that I'm going to make. So this is the number of combinations where I get 4 out of the 5. So what does 5 choose 4? | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
And so how many ways can you, if I have 5 baskets, how many ways can I pick 4 of them to be the ones that I make if I'm once again the god of probability? So this is going to be 0.8 to the 4th times 0.2 times out of 5 baskets, I'm choosing 4 that I'm going to make. So this is the number of combinations where I get 4 out of the 5. So what does 5 choose 4? That's 5 factorial over 4 factorial times 1 factorial. Well, that equals just 5. You can work that out. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
So what does 5 choose 4? That's 5 factorial over 4 factorial times 1 factorial. Well, that equals just 5. You can work that out. So it's going to be, so let's just figure this out. So it's going to be 0.8 times 0.8 times 0.8. That's 3 times 0.8. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
You can work that out. So it's going to be, so let's just figure this out. So it's going to be 0.8 times 0.8 times 0.8. That's 3 times 0.8. That equals, did I do that right? Let's see, 0.1, 0.8, 0.8 times 0.8. Yeah, that's right, times 0.2 times 5. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
That's 3 times 0.8. That equals, did I do that right? Let's see, 0.1, 0.8, 0.8 times 0.8. Yeah, that's right, times 0.2 times 5. So 40.96%. So this is 40.96%. So roughly 41% chance that I get exactly 4 out of 5 baskets, which is interesting because that's kind of my free-throw percentage. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
Yeah, that's right, times 0.2 times 5. So 40.96%. So this is 40.96%. So roughly 41% chance that I get exactly 4 out of 5 baskets, which is interesting because that's kind of my free-throw percentage. So it's almost a little less, 2 3rd shot of kind of hitting my free-throw percentage on the mark on that time. And that's probably getting 5 out of 5. Well, there's only one way of getting 5 out of 5. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
So roughly 41% chance that I get exactly 4 out of 5 baskets, which is interesting because that's kind of my free-throw percentage. So it's almost a little less, 2 3rd shot of kind of hitting my free-throw percentage on the mark on that time. And that's probably getting 5 out of 5. Well, there's only one way of getting 5 out of 5. You have to get all 5 of them. So this is 0.8 to the 5th power. Let me get the calculator back. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
Well, there's only one way of getting 5 out of 5. You have to get all 5 of them. So this is 0.8 to the 5th power. Let me get the calculator back. So it's 0.8 times 0.8 times 0.8 times 0.8 equals 0.3276. So 32.77% shot. And then we can add them all up, right? | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
Let me get the calculator back. So it's 0.8 times 0.8 times 0.8 times 0.8 equals 0.3276. So 32.77% shot. And then we can add them all up, right? Because we want the probability of at least 3. So it's going to be that, the probability of getting 5 out of 5, plus the probability of getting a 4 out of 5, which is 0.4096, plus the probability of getting 3 out of 5. So that's 0.2048 equals 0.94208. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
And then we can add them all up, right? Because we want the probability of at least 3. So it's going to be that, the probability of getting 5 out of 5, plus the probability of getting a 4 out of 5, which is 0.4096, plus the probability of getting 3 out of 5. So that's 0.2048 equals 0.94208. So 94.21, roughly, rounding percent chance. Which makes sense. If I have an 80% free-throw percentage on any one shot, I have a very high probability of getting at least 3 out of 5 when I go to the free-throw line. | Probability and combinations (part 2) Probability and Statistics Khan Academy.mp3 |
In the last video, we figured out the mean, variance, and standard deviation for a Bernoulli distribution with specific numbers. What I want to do in this video is to generalize it, to figure out really the formulas for the mean and the variance of a Bernoulli distribution if we don't have the actual numbers. If we just know that the probability of success is p, and the probability of failure is 1 minus p. So let's look at this. Let's look at a population where the probability of success, and we'll define success as 1, as having a probability of p. And the probability of failure is 1 minus p, whatever this might be. And obviously, if you add these two up, if you view them as percentages, these are going to add up to 100%. Or if you add up these two values, you're going to add to 1. And that needs to be the case, because these are the only two possibilities that can occur. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
Let's look at a population where the probability of success, and we'll define success as 1, as having a probability of p. And the probability of failure is 1 minus p, whatever this might be. And obviously, if you add these two up, if you view them as percentages, these are going to add up to 100%. Or if you add up these two values, you're going to add to 1. And that needs to be the case, because these are the only two possibilities that can occur. If this is 60% chance of success, there has to be a 40% chance of failure. 70% chance of success, 30% chance of failure. Now, with this definition of this, and this is the most general definition of a Bernoulli distribution, it's really exactly what we did in the last video. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
And that needs to be the case, because these are the only two possibilities that can occur. If this is 60% chance of success, there has to be a 40% chance of failure. 70% chance of success, 30% chance of failure. Now, with this definition of this, and this is the most general definition of a Bernoulli distribution, it's really exactly what we did in the last video. I now want to calculate the expected value, which is the same thing as the mean of this distribution. And I also want to calculate the variance, which is the same thing as the expected squared distance of a value from the mean. So let's do that. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
Now, with this definition of this, and this is the most general definition of a Bernoulli distribution, it's really exactly what we did in the last video. I now want to calculate the expected value, which is the same thing as the mean of this distribution. And I also want to calculate the variance, which is the same thing as the expected squared distance of a value from the mean. So let's do that. So what is the mean over here? What is going to be the mean? Well, that's just the probability weighted sum of the values that this could take on. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
So let's do that. So what is the mean over here? What is going to be the mean? Well, that's just the probability weighted sum of the values that this could take on. So there is a 1 minus p probability that we get failure, that we get 0. So there's 1 minus p probability of getting 0, so times 0. And then there is a p probability of getting 1, plus p times 1. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
Well, that's just the probability weighted sum of the values that this could take on. So there is a 1 minus p probability that we get failure, that we get 0. So there's 1 minus p probability of getting 0, so times 0. And then there is a p probability of getting 1, plus p times 1. Well, this is pretty easy to calculate. 0 times anything is 0, so that cancels out. And then p times 1 is just going to be p. So pretty straightforward. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
And then there is a p probability of getting 1, plus p times 1. Well, this is pretty easy to calculate. 0 times anything is 0, so that cancels out. And then p times 1 is just going to be p. So pretty straightforward. The mean, the expected value of this distribution is p. And p might be here or something. So once again, it's a value that you cannot actually take on in this distribution, which is interesting. But it is the expected value. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
And then p times 1 is just going to be p. So pretty straightforward. The mean, the expected value of this distribution is p. And p might be here or something. So once again, it's a value that you cannot actually take on in this distribution, which is interesting. But it is the expected value. Now, what is going to be the variance? What is the variance of this distribution? Remember, that is the weighted sum of the squared distances from the mean. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
But it is the expected value. Now, what is going to be the variance? What is the variance of this distribution? Remember, that is the weighted sum of the squared distances from the mean. Now, what's the probability that we get a 0? We already figured that out. There's a 1 minus p probability that we get a 0. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
Remember, that is the weighted sum of the squared distances from the mean. Now, what's the probability that we get a 0? We already figured that out. There's a 1 minus p probability that we get a 0. So that is the probability part. And what is the squared distance from 0 to our mean? Well, the squared distance from 0 to our mean, let me write it over here, is going to be 0. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
There's a 1 minus p probability that we get a 0. So that is the probability part. And what is the squared distance from 0 to our mean? Well, the squared distance from 0 to our mean, let me write it over here, is going to be 0. That's the value we're taking on. Let me do that in blue, since I already wrote 0. 0 minus our mean. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
Well, the squared distance from 0 to our mean, let me write it over here, is going to be 0. That's the value we're taking on. Let me do that in blue, since I already wrote 0. 0 minus our mean. Let me do this in a new color. Minus our mean. That's too similar to that orange. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
0 minus our mean. Let me do this in a new color. Minus our mean. That's too similar to that orange. So I'm going to do the mean in white. 0 minus our mean, which is p, plus the probability that we get a 1, which is just p. This is the squared distance. Let me be very careful. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
That's too similar to that orange. So I'm going to do the mean in white. 0 minus our mean, which is p, plus the probability that we get a 1, which is just p. This is the squared distance. Let me be very careful. It's the probability weighted sum of the squared distances from the mean. Now, what's the distance? Now, we've got a 1. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
Let me be very careful. It's the probability weighted sum of the squared distances from the mean. Now, what's the distance? Now, we've got a 1. And what's the distance between 1 and the mean? It's 1 minus our mean, which is going to be p over here. And we're going to want to square this as well. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
Now, we've got a 1. And what's the distance between 1 and the mean? It's 1 minus our mean, which is going to be p over here. And we're going to want to square this as well. This right here is going to be the variance. Now, let's actually work this out. So this is going to be equal to 1 minus p. Now, 0 minus p is going to be negative p. If you square it, you're just going to get p squared. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
And we're going to want to square this as well. This right here is going to be the variance. Now, let's actually work this out. So this is going to be equal to 1 minus p. Now, 0 minus p is going to be negative p. If you square it, you're just going to get p squared. So it's going to be p squared. Then plus p times. What's 1 minus p squared? | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
So this is going to be equal to 1 minus p. Now, 0 minus p is going to be negative p. If you square it, you're just going to get p squared. So it's going to be p squared. Then plus p times. What's 1 minus p squared? 1 minus p squared is going to be 1 squared, which is just 1. Minus 2 times the product of this. So it's going to be minus 2p. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
What's 1 minus p squared? 1 minus p squared is going to be 1 squared, which is just 1. Minus 2 times the product of this. So it's going to be minus 2p. Let me write it over here. And then plus negative p squared. So plus p squared. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
So it's going to be minus 2p. Let me write it over here. And then plus negative p squared. So plus p squared. Just like that. And now let's multiply everything out. This term right over here is going to be p squared minus p to the third. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
So plus p squared. Just like that. And now let's multiply everything out. This term right over here is going to be p squared minus p to the third. And then this term over here, this whole thing over here, is going to be plus. p times 1 is p. p times negative 2p is negative 2p squared. And then p times p squared is p to the third. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
This term right over here is going to be p squared minus p to the third. And then this term over here, this whole thing over here, is going to be plus. p times 1 is p. p times negative 2p is negative 2p squared. And then p times p squared is p to the third. Now, we can simplify these. p to the third cancels out with p to the third. And then we have p squared minus 2p squared. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
And then p times p squared is p to the third. Now, we can simplify these. p to the third cancels out with p to the third. And then we have p squared minus 2p squared. So this right here becomes, you have this p right over here, so this is equal to p. And then when you add p squared to negative 2p squared, you're left with negative p squared. So minus p squared. And if you want to factor a p out of this, this is going to be equal to p times, if you take p divided by p, you get a 1, p squared divided by p is p. So p times 1 minus p. Which is a pretty neat, clean formula. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
And then we have p squared minus 2p squared. So this right here becomes, you have this p right over here, so this is equal to p. And then when you add p squared to negative 2p squared, you're left with negative p squared. So minus p squared. And if you want to factor a p out of this, this is going to be equal to p times, if you take p divided by p, you get a 1, p squared divided by p is p. So p times 1 minus p. Which is a pretty neat, clean formula. So our variance is p times 1 minus p. And if we want to take it to the next level and figure out the standard deviation, the standard deviation is just the square root of the variance. Which is equal to the square root of p times 1 minus p. And we can even verify that this actually works for the example that we did up here. Our mean is p, the probability of success. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
And if you want to factor a p out of this, this is going to be equal to p times, if you take p divided by p, you get a 1, p squared divided by p is p. So p times 1 minus p. Which is a pretty neat, clean formula. So our variance is p times 1 minus p. And if we want to take it to the next level and figure out the standard deviation, the standard deviation is just the square root of the variance. Which is equal to the square root of p times 1 minus p. And we can even verify that this actually works for the example that we did up here. Our mean is p, the probability of success. We see that it indeed was, it was 0.6. And we know that our variance is essentially the probability of success times the probability of failure. That's our variance right over there. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
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