Sentence stringlengths 102 4.09k | video_title stringlengths 27 104 |
|---|---|
So this, already, we've kind of come up with a neat way of writing the variance. You can essentially take the average of the squares of all of the numbers, in this case a population, and then subtract from that the mean squared of your population. So this could be, depending on how you're calculating things, maybe a sl... | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
So just playing with a little algebra we got from this thing, where you have to, each time, take each of your data points, subtract the mean from it, and then square it. And then, of course, before you had to do anything, you had to calculate the mean. And you take the square, then you sum it all up, then you take the ... | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
And this is, we're getting to something called the raw score method. What we want to do is write this right here, just in terms of xi's. And then we really are at what you call the raw score method, which is oftentimes a faster way of calculating the variance. So let's see, what is mu equal to? What is the mean? The me... | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
So let's see, what is mu equal to? What is the mean? The mean is just equal to the sum from i is equal to 1 to n of each of the terms. You just take the sum of each of the terms, and you divide by the number of terms there are. So that is equal to, so if we look at this thing, this thing can be written as, let me draw ... | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
You just take the sum of each of the terms, and you divide by the number of terms there are. So that is equal to, so if we look at this thing, this thing can be written as, let me draw a line here, this thing can be written as the sum from i is equal to 1 to n of xi squared, all of that over n, minus mu squared. Well, ... | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
So this thing squared. So this thing squared is what? This is x sub i, take the sum to n, i is equal to 1. You're going to square this thing, and then you're going to divide it by, we squared, right? You divide it by n squared. And this might seem like a more, out of all of them, this is actually seems like the simples... | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
You're going to square this thing, and then you're going to divide it by, we squared, right? You divide it by n squared. And this might seem like a more, out of all of them, this is actually seems like the simplest formula for me, where you essentially just take, if you know the mean of your population, you just say, O... | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
But first, I can just take each of the numbers, square them, and then sum them up, and divide by the number of numbers I have. I don't know if I wrote, no, I've erased the last set of numbers, but we could show you that you'll get to the same variance. So to me, this is almost the simplest formula. But this one's even ... | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
But this one's even faster in a lot of ways, because you don't really have to even calculate the mean ahead of time. You can just say, OK, for each xi, I just perform this operation, and then I divide by n squared or n accordingly, and I'll also get to the variance. So you don't have to do this calculation before you f... | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
She was curious if this figure was higher in her city, so she tested. Her null hypothesis is that the proportion in her city is the same as all Americans, 26%. Her alternative hypothesis is it's actually greater than 26%, where P represents the proportion of people in her city that can speak more than one language. She... | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
She found that 40 of 120 people sampled could speak more than one language. So what's going on is, here's the population of her city. She took a sample. Her sample size is 120. And then she calculates her sample proportion, which is 40 out of 120. And this is going to be equal to 1 3rd, which is approximately equal to ... | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
Her sample size is 120. And then she calculates her sample proportion, which is 40 out of 120. And this is going to be equal to 1 3rd, which is approximately equal to 0.33. And then she calculates the test statistic for these results was z is approximately equal to 1.83. We do this in other videos, but just as a remind... | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
And then she calculates the test statistic for these results was z is approximately equal to 1.83. We do this in other videos, but just as a reminder of how she gets this, she's really trying to say, well, how many standard deviations above the assumed proportion, remember, when we're doing the significance test, we're... | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
But that's what this z statistic is, is, well, how many standard deviations above the assumed proportion is that? So the z statistic, and we did this in previous videos, you would find the difference between this, what we got for our sample, our sample proportion, and the assumed true proportion, so 0.33 minus 0.26, al... | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
That is just going to be the assumed proportion, so it would be just this. It'd be the assumed population proportion times one minus the assumed population proportion over n. In this particular situation, that would be 0.26 times one minus 0.26, all of that over our n, that's our sample size, 120. And if you calculate ... | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
So they did all of that for us. And they say, assuming that the necessary conditions are met, they're talking about the necessary conditions to assume that the sampling distribution of the sample proportions is roughly normal, and that's the random condition, the normal condition, the independence condition that we hav... | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
Well, this p-value, this is the p-value would be equal to the probability of in a normal distribution, we're assuming that the sampling distribution is normal because we met the necessary conditions. So in a normal distribution, what is the probability of getting a z greater than or equal to 1.83? So to help us visuali... | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
The mean of the sampling distribution right over here would be the assumed population proportion. So that would be p-naught, when we put that little zero there, that means the assumed population proportion from the null hypothesis, and that's 0.26. And this result that we got from our sample is 1.83 standard deviations... | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
So that would be 1.83 standard deviations. And so what we wanna do, this probability is this area under our normal curve right over here. So now let's get our z table. So notice, this z table gives us the area to the left of a certain z value. We wanted it to the right of a certain z value, but a normal distribution is... | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
So notice, this z table gives us the area to the left of a certain z value. We wanted it to the right of a certain z value, but a normal distribution is symmetric, so instead of saying anything greater than or equal to 1.83 standard deviations above the mean, we could say anything less than or equal to 1.83 standard de... | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
And so we could look at that on this z table right over here. Negative one, let me, negative 1.8, negative 1.83 is this right over here. So 0.0336. So there we have it. So this is approximately 0.0336, 0.0336, or a little over 3% or a little less than 4%. And so what Faye would then do is compare that to the significan... | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
Let's say you're in the babysitting business and you like to keep a log of whom you are babysitting. So in the last month, you babysat six children and you wrote the ages of all six children in your log. But then when you go back to your log, you notice that some blue ink spilled over one of the ages and you forgot how... | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
And at first you're really worried, your whole system of keeping records seems to, you know, you've lost information. But then you remember that every time you wrote down a new age that month, you recalculated the mean. And so you have the mean here of being four, the mean age is four for the six children. So given tha... | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
So given that, given that you know the mean and that you know five out of six of the ages, can you figure out what the sixth age is? And I encourage you to pause the video and try to figure it out on your own. So assuming you've had a shot at it, so let's just call this missing age, let's call that question mark. So le... | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
So let's just think about how do we calculate, how would we calculate a mean if we knew what question mark is? Well, we would take the total, we would take the total of ages, of ages, we would then divide that by the number of children, we would then divide that by the number of ages that we had, and then that would be... | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
Mean times, and I'll just write times the number, times number of data points, or number of ages. So maybe we can use this information, because we're just going to have this missing question mark here, and we know the mean and we know the number of ages, so we just have to solve for the question mark, so let's do that.... | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
The total of ages, that's going to be five plus two plus question mark, plus question mark, plus two, this two, plus two, plus four, plus eight. We're going to divide by the number of ages. We're going to divide it by the number of ages. Well, we have six ages here. One, two, three, four, five, six. Six ages, and that'... | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
Well, we have six ages here. One, two, three, four, five, six. Six ages, and that's going to be equal to the mean. This is going to be equal to the mean. The mean here is four. This is just how you calculate the mean. Let's see if we can simplify this. | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
This is going to be equal to the mean. The mean here is four. This is just how you calculate the mean. Let's see if we can simplify this. Five plus two is seven. Let me do this, that's the wrong color. Five plus two is seven. | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
Let's see if we can simplify this. Five plus two is seven. Let me do this, that's the wrong color. Five plus two is seven. Two plus four is six, plus eight is 14. 14, and then seven plus 14 is 21. We're left with 21 plus question mark over six is equal to four. | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
Five plus two is seven. Two plus four is six, plus eight is 14. 14, and then seven plus 14 is 21. We're left with 21 plus question mark over six is equal to four. Now we can do what we did when we just wrote it all out. We can multiply both sides times the number of ages, the number of data points we have. We can multi... | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
We're left with 21 plus question mark over six is equal to four. Now we can do what we did when we just wrote it all out. We can multiply both sides times the number of ages, the number of data points we have. We can multiply both sides times six. We can multiply both sides, both sides times six. Six on that side, six ... | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
We can multiply both sides times six. We can multiply both sides, both sides times six. Six on that side, six on this side. Six in the numerator, six in the denominator, those cancel. All we're left is, on the left-hand side, we're left with 21 plus question mark. All of these other green numbers, those are simplified,... | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
Six in the numerator, six in the denominator, those cancel. All we're left is, on the left-hand side, we're left with 21 plus question mark. All of these other green numbers, those are simplified, five plus two plus two plus four plus eight is 21, and we still have the question mark. We get 21 plus question mark. I'm g... | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
We get 21 plus question mark. I'm going to do that green color. 21 plus this question mark. The thing that we're trying to solve for, the missing number, is going to be equal to, is going to be equal to four times six. What's four times six? That's 24. What's the question mark? | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
The thing that we're trying to solve for, the missing number, is going to be equal to, is going to be equal to four times six. What's four times six? That's 24. What's the question mark? 21 plus what is equal to 24? We can, of course, you might just, well, it's going to be three, or if you want to, you could say, well,... | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
What's the question mark? 21 plus what is equal to 24? We can, of course, you might just, well, it's going to be three, or if you want to, you could say, well, question mark is going to be, question mark is going to be equal to, is going to be equal to 24 minus 21, which is, of course, three. Which, of course, let me j... | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
Who knows what that's in some blood pressure units. Construct a 95% confidence interval for the true expected blood pressure increase for all patients in a population. So there's some population distribution here. It's a reasonable assumption to think that it is normal. It's a biological process. So if you gave this dr... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
It's a reasonable assumption to think that it is normal. It's a biological process. So if you gave this drug to every person who has ever lived, that will result in some mean increase in blood pressure. Or who knows, maybe it's actually a decrease. And there's also going to be some standard deviation here. It is a norm... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
Or who knows, maybe it's actually a decrease. And there's also going to be some standard deviation here. It is a normal distribution. And the reason why it's reasonable to assume that it's a normal distribution is because it's a biological process. It's going to be the sum of many thousands and millions of random event... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And the reason why it's reasonable to assume that it's a normal distribution is because it's a biological process. It's going to be the sum of many thousands and millions of random events. And things that are sums of many millions and thousands of random events tend to be normal distribution. So this is a population di... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So this is a population distribution. This is the population distribution. And we don't know anything really about it outside of the sample that we have here. Now, what we can do is, and this tends to be a good thing to do when you do have a sample, is just figure out everything that you can figure out about that sampl... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
Now, what we can do is, and this tends to be a good thing to do when you do have a sample, is just figure out everything that you can figure out about that sample from the get go. So we have our seven data points. And you can add them up and divide by 7 and get your sample mean. So our sample mean here is 2.34. And the... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So our sample mean here is 2.34. And then you can also calculate your sample standard deviation. Find the square distance from each of these points to your sample mean, add them up, divide by n minus 1 because it's a sample, then take the square root, and you get your sample standard deviation. And I did this ahead of ... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And I did this ahead of time just to save time. Sample standard deviation is 1.04. And we don't know anything about the population distribution. The thing that we've been doing from the get go is estimating that character with our sample standard deviation. So we've been estimating the true standard deviation of the po... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
The thing that we've been doing from the get go is estimating that character with our sample standard deviation. So we've been estimating the true standard deviation of the population with our sample standard deviation. Now, in this problem, this exact problem, we're going to run into a problem. We're estimating our st... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
We're estimating our standard deviation with an n of only 7. So this is probably going to be a not so good estimate. Let me just write, because n is small. In general, this is considered a bad estimate if n is less than 30. Above 30, you're dealing in the realm of pretty good estimates. And so the whole focus of this v... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
In general, this is considered a bad estimate if n is less than 30. Above 30, you're dealing in the realm of pretty good estimates. And so the whole focus of this video is when we think about the sampling distribution, which is what we're going to use to generate our interval, instead of assuming that the sampling dist... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
We're not going to assume it's a normal distribution, because this is a bad estimate. We're going to assume that it's something called a t distribution. And the t distribution is essentially, the best way to think about it is it's almost engineered. It's almost engineered, so it gives a better estimate of your confiden... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
It's almost engineered, so it gives a better estimate of your confidence intervals and all of that when you do have a small sample size. And it looks very similar to a normal distribution. It has some mean. So this is your mean of your sampling distribution still. But it also has fatter tails. And the way I think about... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So this is your mean of your sampling distribution still. But it also has fatter tails. And the way I think about why it has fatter tails is when you make an assumption that this is the standard deviation for, well, let me take one more step. So normally what we do is we find the estimate of the true standard deviation... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So normally what we do is we find the estimate of the true standard deviation. And then we say that the standard deviation of the sampling distribution is equal to the true standard deviation of our population divided by the square root of n. In this case, n is equal to 7. And then we say, OK, we never know the true st... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
Sometimes you do know. We seldom know the true standard deviation. So if we don't know that, the best thing we can put in there is our sample standard deviation. So the best thing we can put in there is our sample standard deviation. And this right here, this is the whole reason why we don't say that this is just a 95 ... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So the best thing we can put in there is our sample standard deviation. And this right here, this is the whole reason why we don't say that this is just a 95 probability interval. This is the whole reason why we call it a confidence interval, because we're making some assumptions here. This thing is going to change fro... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
This thing is going to change from sample to sample. And in particular, this is going to be a particularly bad estimate when we have a small sample size, a size less than 30. So when you are estimating the standard deviation where you don't know it, you're estimating it with your sample standard deviation. And your sam... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And your sample size is small. And you're going to use this to estimate the standard deviation of your sampling distribution. You don't assume your sampling distribution is a normal distribution. You assume it has fatter tails. And it has fatter tails, because you're essentially underestimating the standard deviation o... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
You assume it has fatter tails. And it has fatter tails, because you're essentially underestimating the standard deviation over here. Anyway, with all of that said, let's just actually go through this problem. So we need to think about a 95% confidence interval around this mean right over here. So a 95% confidence inte... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So we need to think about a 95% confidence interval around this mean right over here. So a 95% confidence interval, if this was a normal distribution, you would just look it up in a z table. But it's not. This is a t distribution. This is a t distribution. We're looking for a 95% confidence interval. So some interval a... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
This is a t distribution. This is a t distribution. We're looking for a 95% confidence interval. So some interval around the mean that encapsulates 95% of the area. For t distribution, you use a t table. And I have a t table ahead of time right over here. And what you want to do is use the two-sided. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So some interval around the mean that encapsulates 95% of the area. For t distribution, you use a t table. And I have a t table ahead of time right over here. And what you want to do is use the two-sided. You want to use a two-sided row for what we're doing right over here. And the best way to think about it is that we... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And what you want to do is use the two-sided. You want to use a two-sided row for what we're doing right over here. And the best way to think about it is that we're symmetric around the mean. And that's why they call it two-sided. One-sided if it was kind of a cumulative percentage up to some critical threshold. But in... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And that's why they call it two-sided. One-sided if it was kind of a cumulative percentage up to some critical threshold. But in this case, it's two-sided. We're symmetric. Or another way to think about it is we're excluding the two sides. So we want the 95% in the middle. And this is a sampling distribution of the sam... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
We're symmetric. Or another way to think about it is we're excluding the two sides. So we want the 95% in the middle. And this is a sampling distribution of the sample mean for n is equal to 7. And I won't go into the details here. But when n is equal to 7, you have 6 degrees of freedom, or n minus 1. And the way that ... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And this is a sampling distribution of the sample mean for n is equal to 7. And I won't go into the details here. But when n is equal to 7, you have 6 degrees of freedom, or n minus 1. And the way that t tables are set up, you go and find the degrees of freedom. So you don't go to the n. You go to the n minus 1. So you... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And the way that t tables are set up, you go and find the degrees of freedom. So you don't go to the n. You go to the n minus 1. So you go to the 6 right here. So if you want to encapsulate 95% of this right over here, and you have an n of 6, you have to go 2.447 standard deviations in each direction. And this t table ... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So if you want to encapsulate 95% of this right over here, and you have an n of 6, you have to go 2.447 standard deviations in each direction. And this t table assumes that you are approximating that standard deviation using your sample standard deviation. So it's another way to think of it. You have to go 2.447 of the... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
You have to go 2.447 of these approximated standard deviations. So let me go right here. So you have to go 2.447. This distance right here is 2.447 times this approximated standard deviation. And sometimes, you'll see this in some statistics books, this thing right here, this exact number, is shown like this. They put ... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
This distance right here is 2.447 times this approximated standard deviation. And sometimes, you'll see this in some statistics books, this thing right here, this exact number, is shown like this. They put a little hat on top of the standard deviation to show that it has been approximated using the sample standard devi... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So we'll put a little hat over here, because frankly, this is the only thing that we can calculate. So this is how far you have to go in each direction. And we know what this value is. We know what the sample distribution is. So let's get our calculator out. So we know our sample standard deviation is 1.04. And we want... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
We know what the sample distribution is. So let's get our calculator out. So we know our sample standard deviation is 1.04. And we want to divide that by the square root of 7. So we get 0.39. So this right here is 0.39. And so if we want to find the distance around this population mean that encapsulates 95% of the popu... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And we want to divide that by the square root of 7. So we get 0.39. So this right here is 0.39. And so if we want to find the distance around this population mean that encapsulates 95% of the population, or of the sampling distribution, we have to multiply 0.39 times 2.447. So let's do that. So times 2.447 is equal to ... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And so if we want to find the distance around this population mean that encapsulates 95% of the population, or of the sampling distribution, we have to multiply 0.39 times 2.447. So let's do that. So times 2.447 is equal to 0.96. So this distance right here is 0.96. And then this distance right here is 0.96. So if you ... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So this distance right here is 0.96. And then this distance right here is 0.96. So if you take a random sample, and that's exactly what we did when we found these seven samples. When we took these seven samples and took their mean, that mean can be viewed as a random sample from the sampling distribution. And so the pr... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
When we took these seven samples and took their mean, that mean can be viewed as a random sample from the sampling distribution. And so the probability, so we can view it. We can say that there's a 95% chance. And we have to actually caveat everything with a confident, because we're doing all of these estimations here.... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And we have to actually caveat everything with a confident, because we're doing all of these estimations here. So it's not a true, precise 95% chance. We're just confident that there's a 95% chance that our random sampling mean right here, so that 2.34, which we can kind of use. We just pick that 2.34 from this distrib... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
We just pick that 2.34 from this distribution right here. So there's a 95% chance that 2.34 is within 0.96 of the true sampling distribution mean, which we know is also the same thing as the population mean. So I'll just say of the population mean. Or we can just rearrange the sentence and say that there is a 95% chanc... | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So there's four suits, each of them have nine cards, so that gives us 36 unique cards. A hand is a collection of nine cards, which can be sorted however the player chooses. So they're essentially telling us that order does not matter. What is the probability of getting all four of the ones? So they want to know the pro... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
What is the probability of getting all four of the ones? So they want to know the probability of getting all four of the ones. So all four ones in my hand of nine. Now this is kind of daunting at first. You're like, gee, I have nine cards and I'm kicking them out of 36. I have to figure out how do I get all of the ones... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Now this is kind of daunting at first. You're like, gee, I have nine cards and I'm kicking them out of 36. I have to figure out how do I get all of the ones. But if we think about it in very simple terms, all a probability is saying is the number of events, or I guess you could say the number of ways in which this acti... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
But if we think about it in very simple terms, all a probability is saying is the number of events, or I guess you could say the number of ways in which this action or this event happens. So this is what the definition of the probability is. It's going to be the number of ways in which event can happen, and when we tal... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
That's the event. And all of these different ways, that's sometimes called the event space. But we actually want to count how many ways that if I get a hand of nine, picking from 36, that I can get the four ones in it, so this is the number of ways in which my event can happen. And we want to divide that into all of th... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And we want to divide that into all of the possibilities. Or maybe I should write it this way. The total number of hands that I can get. So the numerator in blue is the number of hands, or the number of different hands, where I have the four ones, and we're dividing it to divide the total number of hands. Now let's fig... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So the numerator in blue is the number of hands, or the number of different hands, where I have the four ones, and we're dividing it to divide the total number of hands. Now let's figure out the total number of hands first, because at some level this might be more intuitive, and we've actually done this before. Now, th... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And we've done this many, many times. Let me write this. Total number of hands, or total number of possible hands. That's equal to, you can imagine you have nine cards to pick from. The first card you pick is going to be one of 36 cards, then the next one's going to be one of 35, then the next one's going to be one of ... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
That's equal to, you can imagine you have nine cards to pick from. The first card you pick is going to be one of 36 cards, then the next one's going to be one of 35, then the next one's going to be one of 34, 33, 32, 31. We're going to do this nine times, 1, 2, 3, 4, 5, 6, 7, 8, and 9. So that would be the total number... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So that would be the total number of hands if order mattered. But we know, and we've gone over this before, that we don't care about the order. All we care about the actual cards that are in there. So we're over counting here. We're over counting for all of the different rearrangements that these cards would have. It d... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So we're over counting here. We're over counting for all of the different rearrangements that these cards would have. It doesn't matter whether the ace of diamonds is the first card I pick or the last card I pick. The way I've counted them right now, we are counting those as two separate hands. But they aren't two sepa... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
The way I've counted them right now, we are counting those as two separate hands. But they aren't two separate hands, so order doesn't matter. So what we have to do is, we have to divide this by the number of ways you can arrange nine things. So you could put nine of the things in the first position, then eight in the ... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So you could put nine of the things in the first position, then eight in the second, seven in the third, so forth and so on. It essentially becomes 9 factorial, times 2, times 1. And we've seen this multiple times. This is essentially 36 choose 9. This expression right here is the same thing, just so you can relate it ... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
This is essentially 36 choose 9. This expression right here is the same thing, just so you can relate it to the, I guess, combinatorics formulas that you might be familiar with. This is the same thing as 36 factorial over 36 minus 9 factorial, that's what this orange part is over here, divided by 9 factorial, or over 9... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
What's green is what's green, and what is orange is what's orange there. So that's the total number of hands. Now, a little bit more of a nuanced thought process is, how do we figure out the number of ways in which the event can happen, in which we can have all four 1's? So let's figure that out. So number of ways, or ... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So let's figure that out. So number of ways, or maybe we should say this, number of hands with four 1's. And just as a little bit of thought experiment, imagine if we were only taking four cards, if a hand only had four cards in it. Well, if a card only had four hands, if a hand only had four cards in it, then the numb... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Well, if a card only had four hands, if a hand only had four cards in it, then the number of ways to get a hand with four 1's, there would only be one way, one combination. You just have four 1's. That's the only combination with four 1's if we were only picking four cards. But here, we're not only picking four cards. ... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
But here, we're not only picking four cards. Four of the cards are going to be 1's. So four of the cards are going to be 1's. I mean, 1, 2, 3, 4, but the other five cards are going to be different. So 1, 2, 3, 4, 5. So for the other five cards, if you imagine this slot, considering that of the 36, we would have to pick... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
I mean, 1, 2, 3, 4, but the other five cards are going to be different. So 1, 2, 3, 4, 5. So for the other five cards, if you imagine this slot, considering that of the 36, we would have to pick four of them already in order for us to have four 1's. So there's another, well, we've used up four of them, so there's 32 po... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So there's another, well, we've used up four of them, so there's 32 possible cards over in that position of the hand. And then there'd be 31 in that position of the hand. And then there'd be 30, because every time we're picking a card, we're using it up. And now we only have 30 to pick from. Then we only have 29 to pic... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And now we only have 30 to pick from. Then we only have 29 to pick from. And then we have 28 to pick from. And just like we did before, we don't care about order. We don't care if we pick the five of clubs first or whether we pick the five of clubs last. So we shouldn't double count it. So we have to divide by the diff... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And just like we did before, we don't care about order. We don't care if we pick the five of clubs first or whether we pick the five of clubs last. So we shouldn't double count it. So we have to divide by the different number of ways that five cards can be arranged. So we have to divide this by the different ways that ... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So we have to divide by the different number of ways that five cards can be arranged. So we have to divide this by the different ways that five cards can be arranged. So the first card or the first position could be any one of five cards, then four cards, then three cards, then two cards, then one cards. So the number ... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So the number of hands with four 1's is actually just this number. You're actually looking at all of the different ways you can fill up the remaining cards. These four 1's are just going to be four 1's. There's only one way to get that. It's the remaining cards that's going to give all of the different combinations of ... | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.