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That is how many possible three-letter words we can have for the English language if we didn't care about how pronounceable they are, if they meant anything, and if we repeated letters. Now let's ask a different question. What if we said, how many possible three-letter words are there if we want all different letters? So we want all different letters. So these all have to be different letters. Different, different letters. And once again, pause the video and see if you can think it through. | Possible three letter words Probability and Statistics Khan Academy.mp3 |
So we want all different letters. So these all have to be different letters. Different, different letters. And once again, pause the video and see if you can think it through. All right. So this is where permutations start to be useful. Although I, you know, I think a lot of things like this, it's always best to reason through than to try to figure out whether some formula applies to it. | Possible three letter words Probability and Statistics Khan Academy.mp3 |
And once again, pause the video and see if you can think it through. All right. So this is where permutations start to be useful. Although I, you know, I think a lot of things like this, it's always best to reason through than to try to figure out whether some formula applies to it. So in this situation, well, if we went in order, we could have 26 different letters for the first one, 26 different possibilities for the first one. You know, I'm always starting with that one, but there's nothing special about the one on the left. We could say that the one on the right, there's 26 possibilities. | Possible three letter words Probability and Statistics Khan Academy.mp3 |
Although I, you know, I think a lot of things like this, it's always best to reason through than to try to figure out whether some formula applies to it. So in this situation, well, if we went in order, we could have 26 different letters for the first one, 26 different possibilities for the first one. You know, I'm always starting with that one, but there's nothing special about the one on the left. We could say that the one on the right, there's 26 possibilities. Well, for each of those possibilities, for each of those 26 possibilities, there might be 25 possibilities for what we put in the middle one if we say we're gonna figure out the middle one next. And then for each of these 25 times 26 possibilities for where we figured out two of the letters, there's 24 possibilities because we've already used two letters for the last bucket that we haven't filled. And the only reason why I went 26, 25, 24 is to show you there's nothing special about always filling in the leftmost letter or the leftmost chair first. | Possible three letter words Probability and Statistics Khan Academy.mp3 |
We could say that the one on the right, there's 26 possibilities. Well, for each of those possibilities, for each of those 26 possibilities, there might be 25 possibilities for what we put in the middle one if we say we're gonna figure out the middle one next. And then for each of these 25 times 26 possibilities for where we figured out two of the letters, there's 24 possibilities because we've already used two letters for the last bucket that we haven't filled. And the only reason why I went 26, 25, 24 is to show you there's nothing special about always filling in the leftmost letter or the leftmost chair first. It's just about, well, let's just think in terms of, let's fill out one of the buckets first. Hey, we have the most possibilities for that. Once we use something up, then for each of those possibilities, we'll have one less for the next bucket. | Possible three letter words Probability and Statistics Khan Academy.mp3 |
And the only reason why I went 26, 25, 24 is to show you there's nothing special about always filling in the leftmost letter or the leftmost chair first. It's just about, well, let's just think in terms of, let's fill out one of the buckets first. Hey, we have the most possibilities for that. Once we use something up, then for each of those possibilities, we'll have one less for the next bucket. And so I could do 24 times 25 times 26, but just so I don't fully confuse you, I'll go back to what I have been doing. 26 possibilities for the leftmost one. For each of those, you would have 25 possibilities for the next one that you're going to try to figure out because you've already used one letter and they have to be different. | Possible three letter words Probability and Statistics Khan Academy.mp3 |
Once we use something up, then for each of those possibilities, we'll have one less for the next bucket. And so I could do 24 times 25 times 26, but just so I don't fully confuse you, I'll go back to what I have been doing. 26 possibilities for the leftmost one. For each of those, you would have 25 possibilities for the next one that you're going to try to figure out because you've already used one letter and they have to be different. And then for the last bucket, you're going to have 24 possibilities. So this is going to be 26 times 25 times 24, whatever that happens to be. And if we wanted to write it in the notation of permutations, we would say that this is equal to, we're taking 26 things, sorry, not 2p, 20, my brain is malfunctioning, 26, we're figuring out how many permutations are there for putting 26 different things into three different spaces. | Possible three letter words Probability and Statistics Khan Academy.mp3 |
For each of those, you would have 25 possibilities for the next one that you're going to try to figure out because you've already used one letter and they have to be different. And then for the last bucket, you're going to have 24 possibilities. So this is going to be 26 times 25 times 24, whatever that happens to be. And if we wanted to write it in the notation of permutations, we would say that this is equal to, we're taking 26 things, sorry, not 2p, 20, my brain is malfunctioning, 26, we're figuring out how many permutations are there for putting 26 different things into three different spaces. And this is 26 if we just blindly apply the formula, which I never suggest doing. It'd be 26 factorial over 26 minus three factorial, which would be 26 factorial over 23 factorial, which is going to be exactly this right over here because the 23 times 22 times 21 all the way down to one is going to cancel with the 23 factorial. And so the whole point of this video, there's two points, is one, as soon as someone says, oh, how many different letters could you form or something like that, you just don't blindly do permutations or combinations. | Possible three letter words Probability and Statistics Khan Academy.mp3 |
And if we wanted to write it in the notation of permutations, we would say that this is equal to, we're taking 26 things, sorry, not 2p, 20, my brain is malfunctioning, 26, we're figuring out how many permutations are there for putting 26 different things into three different spaces. And this is 26 if we just blindly apply the formula, which I never suggest doing. It'd be 26 factorial over 26 minus three factorial, which would be 26 factorial over 23 factorial, which is going to be exactly this right over here because the 23 times 22 times 21 all the way down to one is going to cancel with the 23 factorial. And so the whole point of this video, there's two points, is one, as soon as someone says, oh, how many different letters could you form or something like that, you just don't blindly do permutations or combinations. You think about, well, what is being asked in the question here? I really just have to take 26 times 26 times 26. The other thing I want to point out, and I know I keep pointing it out and it's probably getting tiring to you, is even when permutations are applicable, in my brain at least, it's always more valuable to just try to reason through the problem as opposed to just saying, oh, there's this formula that I remember from weeks or years ago in my life that had n factorial and k factorial and I have to memorize it, I have to look it up, always much more useful to just reason it through. | Possible three letter words Probability and Statistics Khan Academy.mp3 |
A random sample of 200 computers show that 12 computers have the defect. What critical value, z star, should Elena use to construct this confidence interval? So before I even ask you to pause this video, let me just give you a little reminder of what a critical value is. Remember, the whole point behind confidence intervals are we have some true population parameter. In this case, it is the proportion of computers that have a defect. So there's some true population proportion. We don't know what that is, but we try to estimate it. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
Remember, the whole point behind confidence intervals are we have some true population parameter. In this case, it is the proportion of computers that have a defect. So there's some true population proportion. We don't know what that is, but we try to estimate it. We take a sample. In this case, it's a sample, a random sample of 200 computers. We take a random sample, and then we estimate this by calculating the sample proportion. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
We don't know what that is, but we try to estimate it. We take a sample. In this case, it's a sample, a random sample of 200 computers. We take a random sample, and then we estimate this by calculating the sample proportion. But then we also wanna construct a confidence interval. And remember, a confidence interval at a 94% confidence level means that if we were to keep doing this, and if we were to keep creating intervals around these statistics, so maybe that's the confidence interval around that one. Maybe if we were to do it again, that's the confidence interval around that one, that 94%, that roughly, as I keep doing this over and over again, that roughly 94% of these intervals are going to overlap with our true population parameter. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
We take a random sample, and then we estimate this by calculating the sample proportion. But then we also wanna construct a confidence interval. And remember, a confidence interval at a 94% confidence level means that if we were to keep doing this, and if we were to keep creating intervals around these statistics, so maybe that's the confidence interval around that one. Maybe if we were to do it again, that's the confidence interval around that one, that 94%, that roughly, as I keep doing this over and over again, that roughly 94% of these intervals are going to overlap with our true population parameter. And the way that we do this is we take the statistic. Let me just write this in general form, even if we're not talking about a proportion. It could be if we're trying to estimate the population mean. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
Maybe if we were to do it again, that's the confidence interval around that one, that 94%, that roughly, as I keep doing this over and over again, that roughly 94% of these intervals are going to overlap with our true population parameter. And the way that we do this is we take the statistic. Let me just write this in general form, even if we're not talking about a proportion. It could be if we're trying to estimate the population mean. So we take our statistic, statistic, and then we go plus or minus around that statistic, plus or minus around that statistic. And then we say, okay, how many standard deviations for the sampling distribution do we wanna go above or beyond? So the number of standard deviations we wanna go, that is our critical value. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
It could be if we're trying to estimate the population mean. So we take our statistic, statistic, and then we go plus or minus around that statistic, plus or minus around that statistic. And then we say, okay, how many standard deviations for the sampling distribution do we wanna go above or beyond? So the number of standard deviations we wanna go, that is our critical value. And then we multiply that times the standard deviation of the statistic, of the statistic. Now in this particular situation, our statistic is p hat from this one sample that Elena made. So it's that one sample proportion that she was able to calculate, plus or minus z star. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
So the number of standard deviations we wanna go, that is our critical value. And then we multiply that times the standard deviation of the statistic, of the statistic. Now in this particular situation, our statistic is p hat from this one sample that Elena made. So it's that one sample proportion that she was able to calculate, plus or minus z star. And we're gonna think about which z star, because that's essentially the question, the critical value. So plus or minus some critical value times, and what we do, because in order to actually calculate the true standard deviation of the sampling distribution, of the sample proportions, well, then you actually have to know the population parameter. But we don't know that, so we multiply that times the standard error of the statistic. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
So it's that one sample proportion that she was able to calculate, plus or minus z star. And we're gonna think about which z star, because that's essentially the question, the critical value. So plus or minus some critical value times, and what we do, because in order to actually calculate the true standard deviation of the sampling distribution, of the sample proportions, well, then you actually have to know the population parameter. But we don't know that, so we multiply that times the standard error of the statistic. And we've done this in previous videos. But the key question here is, what is our z star? And what we really needed to think about is, assuming that the sampling distribution is roughly normal, and this is the mean of it, which would actually be our true population parameter, which we do not know. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
But we don't know that, so we multiply that times the standard error of the statistic. And we've done this in previous videos. But the key question here is, what is our z star? And what we really needed to think about is, assuming that the sampling distribution is roughly normal, and this is the mean of it, which would actually be our true population parameter, which we do not know. But how many standard deviations above and below the mean in order to capture 94% of the probability? 94% of the area. So this distance right over here, where this is 94%, this number of standard deviations, that is z star right over here. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
And what we really needed to think about is, assuming that the sampling distribution is roughly normal, and this is the mean of it, which would actually be our true population parameter, which we do not know. But how many standard deviations above and below the mean in order to capture 94% of the probability? 94% of the area. So this distance right over here, where this is 94%, this number of standard deviations, that is z star right over here. Now, all we have to really do is look it up on a z table, but even there we have to be careful. And you should always be careful which type of z table you're using, or if you're using a calculator function, what your calculator function does. Because a lot of z tables will actually do something like this. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
So this distance right over here, where this is 94%, this number of standard deviations, that is z star right over here. Now, all we have to really do is look it up on a z table, but even there we have to be careful. And you should always be careful which type of z table you're using, or if you're using a calculator function, what your calculator function does. Because a lot of z tables will actually do something like this. For a given z, they'll say, what is the total area going all the way from negative infinity up to including z standard deviations above the mean? So when you look up a lot of z tables, they will give you this area. So one way to think about this, we wanna find the critical value, we wanna find the z that leaves not 6% unshaded in, but leaves 3% unshaded in. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
Because a lot of z tables will actually do something like this. For a given z, they'll say, what is the total area going all the way from negative infinity up to including z standard deviations above the mean? So when you look up a lot of z tables, they will give you this area. So one way to think about this, we wanna find the critical value, we wanna find the z that leaves not 6% unshaded in, but leaves 3% unshaded in. Where did I get that from? Well, 100% minus 94% is 6%. But remember, this is going to be symmetric on the left and the right. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
So one way to think about this, we wanna find the critical value, we wanna find the z that leaves not 6% unshaded in, but leaves 3% unshaded in. Where did I get that from? Well, 100% minus 94% is 6%. But remember, this is going to be symmetric on the left and the right. So you're gonna want 3% not shaded in over here, and 3% not shaded in over here. So when I look at a traditional z table that is viewing it from this point of view, this cumulative area, what I really wanna do is find the z that is leaving 3% open over here, which would mean the z that is filling in 97% over here, not 94%. But if I find this z, but if I were to stop it right over there as well, then I would have 3% available there, and then the true area that we're filling in would be 94%. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
But remember, this is going to be symmetric on the left and the right. So you're gonna want 3% not shaded in over here, and 3% not shaded in over here. So when I look at a traditional z table that is viewing it from this point of view, this cumulative area, what I really wanna do is find the z that is leaving 3% open over here, which would mean the z that is filling in 97% over here, not 94%. But if I find this z, but if I were to stop it right over there as well, then I would have 3% available there, and then the true area that we're filling in would be 94%. So with that out of the way, let's look that up. What z gives us, fills us, fills in 97% of the area? So I got a z table. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
But if I find this z, but if I were to stop it right over there as well, then I would have 3% available there, and then the true area that we're filling in would be 94%. So with that out of the way, let's look that up. What z gives us, fills us, fills in 97% of the area? So I got a z table. This is actually the one that you would see if you were, say, taking AP Statistics. And we would just look up, where do we get to 97%? And so it is 97%, looks like it is right about here. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
So I got a z table. This is actually the one that you would see if you were, say, taking AP Statistics. And we would just look up, where do we get to 97%? And so it is 97%, looks like it is right about here. That looks like the closest number. This is 6 10,000ths above it. This is only 1 10,000th below it. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
And so it is 97%, looks like it is right about here. That looks like the closest number. This is 6 10,000ths above it. This is only 1 10,000th below it. And so this is, let's see, you would look at the row first. If we look at the row, it is 1.8, 1.88 is our z. So going back to this right over here, if our z is equal to 1.88, so this is equal to 1.88, then all of this area, up to and including 1.88 standard deviations above the mean, that would be 97%. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
This is only 1 10,000th below it. And so this is, let's see, you would look at the row first. If we look at the row, it is 1.8, 1.88 is our z. So going back to this right over here, if our z is equal to 1.88, so this is equal to 1.88, then all of this area, up to and including 1.88 standard deviations above the mean, that would be 97%. But if you were to go 1.88 standard deviations above the mean and 1.88 standard deviations below the mean, that would leave 3% open on either side. And so this would be 94%. So this would be 94%. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
So going back to this right over here, if our z is equal to 1.88, so this is equal to 1.88, then all of this area, up to and including 1.88 standard deviations above the mean, that would be 97%. But if you were to go 1.88 standard deviations above the mean and 1.88 standard deviations below the mean, that would leave 3% open on either side. And so this would be 94%. So this would be 94%. But to answer their question, what critical value z star? Well, this is going to be 1.88. And we're done. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
I'll assume it's a quarter or something. Let's see. So this is a quarter. Let me draw my best attempt at a profile of George Washington. Well, that'll do. So it's a fair coin, and we're going to flip it a bunch of times and figure out the different probabilities. So let's start with a straightforward one. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
Let me draw my best attempt at a profile of George Washington. Well, that'll do. So it's a fair coin, and we're going to flip it a bunch of times and figure out the different probabilities. So let's start with a straightforward one. Let's just flip it once. So with one flip of the coin, what's the probability of getting heads? Well, there's two equally likely possibilities, and the one with heads is one of those two equally likely possibilities, so there's a 1 half chance. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
So let's start with a straightforward one. Let's just flip it once. So with one flip of the coin, what's the probability of getting heads? Well, there's two equally likely possibilities, and the one with heads is one of those two equally likely possibilities, so there's a 1 half chance. Same thing if we were to ask what is the probability of getting tails. There are two equally likely possibilities, and one of those gives us tails, so 1 half. And this is one thing to realize. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
Well, there's two equally likely possibilities, and the one with heads is one of those two equally likely possibilities, so there's a 1 half chance. Same thing if we were to ask what is the probability of getting tails. There are two equally likely possibilities, and one of those gives us tails, so 1 half. And this is one thing to realize. If you take the probabilities of heads plus the probabilities of tails, you get 1 half plus 1 half, which is 1. And this is generally 2. The sum of the probabilities of all of the possible events should be equal to 1, and that makes sense because you're adding up all of these fractions, and the numerator will then add up to all of the possible events. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
And this is one thing to realize. If you take the probabilities of heads plus the probabilities of tails, you get 1 half plus 1 half, which is 1. And this is generally 2. The sum of the probabilities of all of the possible events should be equal to 1, and that makes sense because you're adding up all of these fractions, and the numerator will then add up to all of the possible events. The denominator is always all of the possible events, so you have all of the possible events over all of the possible events when you add all of these things up. Now let's take it up a notch. Let's figure out the probability of... | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
The sum of the probabilities of all of the possible events should be equal to 1, and that makes sense because you're adding up all of these fractions, and the numerator will then add up to all of the possible events. The denominator is always all of the possible events, so you have all of the possible events over all of the possible events when you add all of these things up. Now let's take it up a notch. Let's figure out the probability of... I'm going to take this coin, and I'm going to flip it twice. The probability of getting heads and then getting another heads. The probability of getting a head and then another head. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
Let's figure out the probability of... I'm going to take this coin, and I'm going to flip it twice. The probability of getting heads and then getting another heads. The probability of getting a head and then another head. So there's two ways to think about it. One way is to just think about all of the different possibilities. I could get a head on the first flip and a head on the second flip, head on the first flip, tail on the second flip. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
The probability of getting a head and then another head. So there's two ways to think about it. One way is to just think about all of the different possibilities. I could get a head on the first flip and a head on the second flip, head on the first flip, tail on the second flip. I could get tails on the first flip, heads on the second flip, or I could get tails on both flips. So there's four distinct, equally likely possibilities. Four distinct, equally likely outcomes here. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
I could get a head on the first flip and a head on the second flip, head on the first flip, tail on the second flip. I could get tails on the first flip, heads on the second flip, or I could get tails on both flips. So there's four distinct, equally likely possibilities. Four distinct, equally likely outcomes here. One way to think about it is on the first flip, I have two possibilities. On the second flip, I have another two possibilities. I could have heads or tails, heads or tails. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
Four distinct, equally likely outcomes here. One way to think about it is on the first flip, I have two possibilities. On the second flip, I have another two possibilities. I could have heads or tails, heads or tails. So I have four possibilities. For each of these possibilities, for each of these two, I have two possibilities here. So either way, I have four equally likely possibilities. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
I could have heads or tails, heads or tails. So I have four possibilities. For each of these possibilities, for each of these two, I have two possibilities here. So either way, I have four equally likely possibilities. How many of those meet our constraints? Well, we have it right over here. This one right over here, having two heads meets our constraints. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
So either way, I have four equally likely possibilities. How many of those meet our constraints? Well, we have it right over here. This one right over here, having two heads meets our constraints. So this is, and there's only one of those possibilities. I've only circled one of the four scenarios. So there's a 1 4th chance of that happening. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
This one right over here, having two heads meets our constraints. So this is, and there's only one of those possibilities. I've only circled one of the four scenarios. So there's a 1 4th chance of that happening. Another way you could think about this, and this is because these are independent events, and this is a very important idea to understand in probability, and we'll also study scenarios that are not independent, but these are independent events. What happens in the first flip in no way affects what happens in the second flip. This is actually one thing that many people don't realize. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
So there's a 1 4th chance of that happening. Another way you could think about this, and this is because these are independent events, and this is a very important idea to understand in probability, and we'll also study scenarios that are not independent, but these are independent events. What happens in the first flip in no way affects what happens in the second flip. This is actually one thing that many people don't realize. There's something called the gambler's fallacy, where someone thinks if I got a bunch of heads in a row, then all of a sudden it becomes more likely on the next flip to get a tail. That is not the case. Every flip is an independent event. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
This is actually one thing that many people don't realize. There's something called the gambler's fallacy, where someone thinks if I got a bunch of heads in a row, then all of a sudden it becomes more likely on the next flip to get a tail. That is not the case. Every flip is an independent event. What happened in the past in these flips does not affect the probabilities going forward. So the probability of getting heads on the first flip in no way, or the fact that you got heads on the first flip, in no way affects that you got heads on the second flip. So if you can make that assumption, you could say that the probability of getting heads and heads, or heads and then heads, is going to be the same thing as the probability of getting heads on the first flip times the probability of getting heads on the second flip. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
Every flip is an independent event. What happened in the past in these flips does not affect the probabilities going forward. So the probability of getting heads on the first flip in no way, or the fact that you got heads on the first flip, in no way affects that you got heads on the second flip. So if you can make that assumption, you could say that the probability of getting heads and heads, or heads and then heads, is going to be the same thing as the probability of getting heads on the first flip times the probability of getting heads on the second flip. We know the probability of getting heads on the first flip is one-half, and the probability of getting heads on the second flip is one-half. And so we have one-half times one-half, which is equal to 1 4th, which is exactly what we got when we tried out all of the different scenarios, all of the equally likely possibilities. Let's take it up another notch. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
So if you can make that assumption, you could say that the probability of getting heads and heads, or heads and then heads, is going to be the same thing as the probability of getting heads on the first flip times the probability of getting heads on the second flip. We know the probability of getting heads on the first flip is one-half, and the probability of getting heads on the second flip is one-half. And so we have one-half times one-half, which is equal to 1 4th, which is exactly what we got when we tried out all of the different scenarios, all of the equally likely possibilities. Let's take it up another notch. Let's figure out the probability, and we've kind of been ignoring tails, so let's pay some attention to tails. The probability of getting tails and then heads and then tails, so this exact series of events. So I'm not saying in any order, two tails and a head. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
Let's take it up another notch. Let's figure out the probability, and we've kind of been ignoring tails, so let's pay some attention to tails. The probability of getting tails and then heads and then tails, so this exact series of events. So I'm not saying in any order, two tails and a head. I'm saying in this exact order, the first flip is a tails, second flip is a heads, and then third flip is a tail. So once again, these are all independent events. The fact that I got tails on the first flip in no way affects the probability of getting a heads on the second flip. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
So I'm not saying in any order, two tails and a head. I'm saying in this exact order, the first flip is a tails, second flip is a heads, and then third flip is a tail. So once again, these are all independent events. The fact that I got tails on the first flip in no way affects the probability of getting a heads on the second flip. And that in no way affects the probability of getting a tails on the third flip. So because these are independent events, we can say this is the same thing as the probability of getting tails on the first flip, times the probability of getting heads on the second flip times the probability of getting tails on the third flip. And we know these are all independent events. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
The fact that I got tails on the first flip in no way affects the probability of getting a heads on the second flip. And that in no way affects the probability of getting a tails on the third flip. So because these are independent events, we can say this is the same thing as the probability of getting tails on the first flip, times the probability of getting heads on the second flip times the probability of getting tails on the third flip. And we know these are all independent events. So this right over here is 1 half times 1 half times 1 half. 1 half times 1 half is 1 fourth. 1 fourth times 1 half is equal to 1 eighth. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
And we know these are all independent events. So this right over here is 1 half times 1 half times 1 half. 1 half times 1 half is 1 fourth. 1 fourth times 1 half is equal to 1 eighth. So this is equal to 1 eighth. And we can verify it. Let's try it all over the different scenarios again. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
1 fourth times 1 half is equal to 1 eighth. So this is equal to 1 eighth. And we can verify it. Let's try it all over the different scenarios again. So you could get heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
Let's try it all over the different scenarios again. So you could get heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads. You could get heads, tails, tails. You can get tails, heads, heads. This is a little tricky sometimes. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
You could get heads, tails, heads. You could get heads, tails, tails. You can get tails, heads, heads. This is a little tricky sometimes. You want to make sure you're being exhaustive in all of the different possibilities here. You could get tails, heads, tails. You could get tails, heads. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
This is a little tricky sometimes. You want to make sure you're being exhaustive in all of the different possibilities here. You could get tails, heads, tails. You could get tails, heads. Or you could get tails, tails, tails. And what we see here is that we got exactly 8 equally likely possibilities. We have 8 equally likely possibilities. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
You could get tails, heads. Or you could get tails, tails, tails. And what we see here is that we got exactly 8 equally likely possibilities. We have 8 equally likely possibilities. And the tail, heads, tails is exactly one of them. It is this possibility right over here. So it is one of 8 equally likely possibilities. | Compound probability of independent events Probability and Statistics Khan Academy.mp3 |
It's all for free. So problem number four, and it's, at least in my mind, pretty good practice. For a standard normal distribution, place the following in order from smallest to largest. So let's see, percentage of data below 1, negative 1. OK, let's draw our standard normal distribution. So a standard normal distribution is one where the mean is, sorry, that's due to the standard deviation, is one where the mean, mu for mean, is where the mean is equal to 0 and the standard deviation is equal to 1. So let me draw that standard normal distribution. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
So let's see, percentage of data below 1, negative 1. OK, let's draw our standard normal distribution. So a standard normal distribution is one where the mean is, sorry, that's due to the standard deviation, is one where the mean, mu for mean, is where the mean is equal to 0 and the standard deviation is equal to 1. So let me draw that standard normal distribution. So let me draw the axis right like that. Let me see if I can draw a nice looking bell curve. There's the bell curve right there. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
So let me draw that standard normal distribution. So let me draw the axis right like that. Let me see if I can draw a nice looking bell curve. There's the bell curve right there. You get the idea. And this is a standard normal distribution, so the mean, or you can kind of view the center point right here. It's not skewed. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
There's the bell curve right there. You get the idea. And this is a standard normal distribution, so the mean, or you can kind of view the center point right here. It's not skewed. This is the mean is going to be 0 right there. And the standard deviation is 1. So if we go one standard deviation to the right, that is going to be 1. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
It's not skewed. This is the mean is going to be 0 right there. And the standard deviation is 1. So if we go one standard deviation to the right, that is going to be 1. If you go two standard deviations, it's going to be 2, three standard deviations, 3, just like that. One standard deviation to the left is going to be minus 1. Two standard deviations to the left will be minus 2, and so on and so forth. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
So if we go one standard deviation to the right, that is going to be 1. If you go two standard deviations, it's going to be 2, three standard deviations, 3, just like that. One standard deviation to the left is going to be minus 1. Two standard deviations to the left will be minus 2, and so on and so forth. Minus 3 will be three standard deviations to the left, because the standard deviation is 1. So let's see if we can answer this question. So what's the percentage of data below 1? | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
Two standard deviations to the left will be minus 2, and so on and so forth. Minus 3 will be three standard deviations to the left, because the standard deviation is 1. So let's see if we can answer this question. So what's the percentage of data below 1? So the percentage, the part A, that's this stuff right here. So everything below 1, so it's all of, well not just that little center portion, it keeps going. Everything below 1, right? | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
So what's the percentage of data below 1? So the percentage, the part A, that's this stuff right here. So everything below 1, so it's all of, well not just that little center portion, it keeps going. Everything below 1, right? Percentage of data below 1. So this is another situation where we should use the empirical rule. Never hurts to get more practice. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
Everything below 1, right? Percentage of data below 1. So this is another situation where we should use the empirical rule. Never hurts to get more practice. Empirical rule, or maybe the better way to remember the empirical rule is just the 68, 95, 99.7 rule. And I call that a better way because it essentially gives you the rule. These are just the numbers that you have to essentially memorize. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
Never hurts to get more practice. Empirical rule, or maybe the better way to remember the empirical rule is just the 68, 95, 99.7 rule. And I call that a better way because it essentially gives you the rule. These are just the numbers that you have to essentially memorize. And if you have a calculator or normal distribution table, you don't have to do this. But sometimes in class, or people want you to estimate percentages, and so it's good to do, you can impress people if you can do this in your head. So let's see if we can use the empirical rule to answer this question, the area under the bell curve all the way up to 1, or everything to the left of 1. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
These are just the numbers that you have to essentially memorize. And if you have a calculator or normal distribution table, you don't have to do this. But sometimes in class, or people want you to estimate percentages, and so it's good to do, you can impress people if you can do this in your head. So let's see if we can use the empirical rule to answer this question, the area under the bell curve all the way up to 1, or everything to the left of 1. So the empirical rule tells us that this middle area between one standard deviation to the left and one standard deviation to the right, that right there is 68%. We saw that in the previous video as well. That's what the empirical rule tells us. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
So let's see if we can use the empirical rule to answer this question, the area under the bell curve all the way up to 1, or everything to the left of 1. So the empirical rule tells us that this middle area between one standard deviation to the left and one standard deviation to the right, that right there is 68%. We saw that in the previous video as well. That's what the empirical rule tells us. Now, if that 68% we saw in the last video, that everything else combined, it all has to add up to 1, or to 100%, that this left-hand tail, let me draw it a little bit, this part right here, plus this part right here, has to add up, when you add it to 68, has to add up to 1, or to 100%. So those two combined are 32%. 32 plus 68 is 100. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
That's what the empirical rule tells us. Now, if that 68% we saw in the last video, that everything else combined, it all has to add up to 1, or to 100%, that this left-hand tail, let me draw it a little bit, this part right here, plus this part right here, has to add up, when you add it to 68, has to add up to 1, or to 100%. So those two combined are 32%. 32 plus 68 is 100. Now, this is symmetrical. These two things are the exact same. So if they add up to 32, this right here is 16%, and this right here is 16%. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
32 plus 68 is 100. Now, this is symmetrical. These two things are the exact same. So if they add up to 32, this right here is 16%, and this right here is 16%. Now, the question, they want us to know the area of everything, let me do it in a new color, everything less than 1. The percentage of data below 1. So everything to the left of this point. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
So if they add up to 32, this right here is 16%, and this right here is 16%. Now, the question, they want us to know the area of everything, let me do it in a new color, everything less than 1. The percentage of data below 1. So everything to the left of this point. So it's the 68%, it's right there, so it's 68%, which is this middle area within one standard deviation, plus this left branch right there. So 68 plus 16%, which is what? That's equal to 84%. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
So everything to the left of this point. So it's the 68%, it's right there, so it's 68%, which is this middle area within one standard deviation, plus this left branch right there. So 68 plus 16%, which is what? That's equal to 84%. So this part A is 84%. They're going to want us to put it in order eventually, but it's good to just solve it. That's really the hard part. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
That's equal to 84%. So this part A is 84%. They're going to want us to put it in order eventually, but it's good to just solve it. That's really the hard part. Once we know the numbers, ordering is pretty easy. Part B. The percentage of data below minus 1. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
That's really the hard part. Once we know the numbers, ordering is pretty easy. Part B. The percentage of data below minus 1. So minus 1 is right there. So they really just want us to figure out this area right here, the percentage of data below minus 1. Well, that's going to be 16%. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
The percentage of data below minus 1. So minus 1 is right there. So they really just want us to figure out this area right here, the percentage of data below minus 1. Well, that's going to be 16%. We just figured that out. And you could have already known, just without even knowing the empirical, just looking at a normal distribution, that this entire area, that part B is a subset of part A, so it's going to be a smaller number. So if you just had to order things, you could have made that intuition, but it's good to do practice with the empirical rule. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
Well, that's going to be 16%. We just figured that out. And you could have already known, just without even knowing the empirical, just looking at a normal distribution, that this entire area, that part B is a subset of part A, so it's going to be a smaller number. So if you just had to order things, you could have made that intuition, but it's good to do practice with the empirical rule. Now, part C. They want to know what's the mean. Well, that's the easiest thing. The mean of a standard normal distribution, by definition, is 0. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
So if you just had to order things, you could have made that intuition, but it's good to do practice with the empirical rule. Now, part C. They want to know what's the mean. Well, that's the easiest thing. The mean of a standard normal distribution, by definition, is 0. So number C is 0. D. The standard deviation. Well, by definition, the standard deviation for the standard normal distribution is 1. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
The mean of a standard normal distribution, by definition, is 0. So number C is 0. D. The standard deviation. Well, by definition, the standard deviation for the standard normal distribution is 1. So this is 1 right here. This is easier than I thought it would be. Part E. The percentage of data above 2. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
Well, by definition, the standard deviation for the standard normal distribution is 1. So this is 1 right here. This is easier than I thought it would be. Part E. The percentage of data above 2. So they want the percentage of data above 2. So we know from the 68, 95, 99.7 rule that if we want to know how much data is within 2 standard deviations. So let me do it in a new color. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
Part E. The percentage of data above 2. So they want the percentage of data above 2. So we know from the 68, 95, 99.7 rule that if we want to know how much data is within 2 standard deviations. So let me do it in a new color. So if we're looking for, from this, let me do it in a more vibrant color, green. If we're looking from this point to this point, so it's within 2 standard deviations, right? The standard deviation here is 1. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
So let me do it in a new color. So if we're looking for, from this, let me do it in a more vibrant color, green. If we're looking from this point to this point, so it's within 2 standard deviations, right? The standard deviation here is 1. If we're looking within 2 standard deviations, that whole area right there, by the empirical rule, is 95% within 2 standard deviations. This is 95%. Which tells us that everything else combined. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
The standard deviation here is 1. If we're looking within 2 standard deviations, that whole area right there, by the empirical rule, is 95% within 2 standard deviations. This is 95%. Which tells us that everything else combined. So if you take this yellow portion right here, and this yellow portion right here. So everything beyond 2 standard deviations in either direction, that has to be the remainder. So everything in the middle was 95 within 2 standard deviations. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
Which tells us that everything else combined. So if you take this yellow portion right here, and this yellow portion right here. So everything beyond 2 standard deviations in either direction, that has to be the remainder. So everything in the middle was 95 within 2 standard deviations. So that has to be 5%. If you add that tail and that tail together, everything to the left and right of 2 standard deviations. Well, I've made the argument before. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
So everything in the middle was 95 within 2 standard deviations. So that has to be 5%. If you add that tail and that tail together, everything to the left and right of 2 standard deviations. Well, I've made the argument before. Everything is symmetrical. This and this are equal. So this right here is 2.5%. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
Well, I've made the argument before. Everything is symmetrical. This and this are equal. So this right here is 2.5%. And this right here is also 2.5%. So they're asking us the percentage of data above 2. That's this tail. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
So this right here is 2.5%. And this right here is also 2.5%. So they're asking us the percentage of data above 2. That's this tail. Just this tail right here. The percentage of data more than 2 standard deviations away from the mean. So that's 2.5%. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
That's this tail. Just this tail right here. The percentage of data more than 2 standard deviations away from the mean. So that's 2.5%. I'll do it in a darker color. 2.5%. Now they're asking us, let's see, place the following in order from smallest to largest. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
So that's 2.5%. I'll do it in a darker color. 2.5%. Now they're asking us, let's see, place the following in order from smallest to largest. So there's a little bit of ambiguity here. Because if they're saying the percentage of data below 1, do they want us to say, well, it's 84%. So should we consider the answer to part A, 84? | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
Now they're asking us, let's see, place the following in order from smallest to largest. So there's a little bit of ambiguity here. Because if they're saying the percentage of data below 1, do they want us to say, well, it's 84%. So should we consider the answer to part A, 84? Or should we consider, if they said the fraction of data below 1, I would say 0.84. So it depends on how they want to interpret it. Same way here. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
So should we consider the answer to part A, 84? Or should we consider, if they said the fraction of data below 1, I would say 0.84. So it depends on how they want to interpret it. Same way here. The percentage of data below minus 1. I could say the answer is 16. 16 is the percentage below minus 1. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
Same way here. The percentage of data below minus 1. I could say the answer is 16. 16 is the percentage below minus 1. But the actual number, if I said the fraction of data below minus 1, I would say 0.16. So this actually would be very different in how we order it. Similarly, if someone asked me the percentage, I'd say, oh, that's 2.5. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
16 is the percentage below minus 1. But the actual number, if I said the fraction of data below minus 1, I would say 0.16. So this actually would be very different in how we order it. Similarly, if someone asked me the percentage, I'd say, oh, that's 2.5. But the actual number is 0.025. That's the actual fraction or the actual decimal. So I mean, this is just ordering numbers. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
Similarly, if someone asked me the percentage, I'd say, oh, that's 2.5. But the actual number is 0.025. That's the actual fraction or the actual decimal. So I mean, this is just ordering numbers. So I shouldn't fixate on this too much. But let's just say that they're going with the decimals. So if we wanted to do it that way, they want to do it from smallest to largest. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
So I mean, this is just ordering numbers. So I shouldn't fixate on this too much. But let's just say that they're going with the decimals. So if we wanted to do it that way, they want to do it from smallest to largest. The smallest number we have here is c. That's 0. And then the next smallest is e, which is 0.025. Then the next smallest is b, which is 0.16. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
So if we wanted to do it that way, they want to do it from smallest to largest. The smallest number we have here is c. That's 0. And then the next smallest is e, which is 0.025. Then the next smallest is b, which is 0.16. And then the next one after that is a, which is 0.84. And then the largest would be the standard deviation, d. So the answer is c bad. And obviously, the order would be different if the answer to this, instead of saying it was 0.84, if you said it was 84, because you're asking for the percentage. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
Then the next smallest is b, which is 0.16. And then the next one after that is a, which is 0.84. And then the largest would be the standard deviation, d. So the answer is c bad. And obviously, the order would be different if the answer to this, instead of saying it was 0.84, if you said it was 84, because you're asking for the percentage. So a little bit of ambiguity. If you had a question like this on the exam, I would clarify that with the teacher. But hopefully you found this useful. | k12.org exercise Standard normal distribution and the empirical Khan Academy.mp3 |
So for example, a 1 and a 1, that's doubles. 2 and a 2, that is doubles. A 3 and a 3, a 4 and a 4, a 5 and a 5, a 6 and a 6. All of those are instances of doubles. So the event in question is rolling doubles on two six-sided dice numbered from 1 to 6. So let's think about all of the possible outcomes. Or another way to think about it, let's think about the sample space here. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
All of those are instances of doubles. So the event in question is rolling doubles on two six-sided dice numbered from 1 to 6. So let's think about all of the possible outcomes. Or another way to think about it, let's think about the sample space here. So what can we roll on the first die? So let me write this as die number 1. What are the possible rolls? | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Or another way to think about it, let's think about the sample space here. So what can we roll on the first die? So let me write this as die number 1. What are the possible rolls? Well, they're numbered from 1 to 6. It's a six-sided die. So I can get a 1, a 2, a 3, a 4, a 5, or a 6. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
What are the possible rolls? Well, they're numbered from 1 to 6. It's a six-sided die. So I can get a 1, a 2, a 3, a 4, a 5, or a 6. Now let's think about the second die. So die number 2. Well, exact same thing. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So I can get a 1, a 2, a 3, a 4, a 5, or a 6. Now let's think about the second die. So die number 2. Well, exact same thing. I could get a 1, a 2, a 3, a 4, a 5, or a 6. Now, given these possible outcomes for each of the die, we can now think of the outcomes for both die. So for example, in this, let me draw a grid here, just to make it a little bit neater. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Well, exact same thing. I could get a 1, a 2, a 3, a 4, a 5, or a 6. Now, given these possible outcomes for each of the die, we can now think of the outcomes for both die. So for example, in this, let me draw a grid here, just to make it a little bit neater. So let me draw a line there, and then a line right over there. Let me draw, actually, several of these, just so that we can really do this a little bit clearer. So let me draw a full grid. | Die rolling probability Probability and combinatorics Precalculus Khan Academy.mp3 |
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