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Every hour, based on this regression, you could, it's not unreasonable to expect 15 points improvement, or at least that's what we're seeing, that's what we're seeing from the regression of the data. So let's look at which of these choices actually describe something like that. The model predicts that the student who s... | Interpreting slope of regression line AP Statistics Khan Academy.mp3 |
No, it definitely doesn't say that. The model predicts that students who didn't study at all will have an average score of 15 points. No, we didn't see that. Students, if you take this, if you believe this model, someone who doesn't study at all would get close to, would get between 35 and 40 points, so like a 37 or a ... | Interpreting slope of regression line AP Statistics Khan Academy.mp3 |
Students, if you take this, if you believe this model, someone who doesn't study at all would get close to, would get between 35 and 40 points, so like a 37 or a 38. So don't like that choice. The model predicts that the score will increase 15 points for each additional hour of study time. Yes, that is exactly what we ... | Interpreting slope of regression line AP Statistics Khan Academy.mp3 |
For the final round, each of them will randomly select a card without replacement that will reveal what the star material must be in their craft. Here are the available cards. So I guess the star material is the primary material they need to use in this competition. Maya and Doug both want to get silk as their star mat... | General multiplication rule example dependent events Probability & combinatorics.mp3 |
Maya and Doug both want to get silk as their star material. Maya will draw first, followed by Doug. What is the probability that neither contestant draws silk? Pause this video and see if you can work through that before we work through this together. All right, now let's work through this together. So the probability ... | General multiplication rule example dependent events Probability & combinatorics.mp3 |
Pause this video and see if you can work through that before we work through this together. All right, now let's work through this together. So the probability that neither contestant draws silk. So that would be, I'll just write it another way, the probability that, I'll write MNS for Maya no silk. So Maya no silk and... | General multiplication rule example dependent events Probability & combinatorics.mp3 |
So that would be, I'll just write it another way, the probability that, I'll write MNS for Maya no silk. So Maya no silk and Doug no silk. That's just another way of saying what is the probability that neither contestant draws silk? And so this is going to be equivalent to the probability that Maya does not get silk, M... | General multiplication rule example dependent events Probability & combinatorics.mp3 |
And so this is going to be equivalent to the probability that Maya does not get silk, Maya no silk, right over here, times the probability that Doug doesn't get silk, given that Maya did not get silk, given Maya no silk. This line right over here, this vertical line, this is shorthand for given. And so let's calculate ... | General multiplication rule example dependent events Probability & combinatorics.mp3 |
So this is going to be equal to the probability that Maya gets no silk, she picked first. There's six options out of here. Five of them are not silk. So it is five over six. And then the probability that Doug does not get silk, given that Maya did not get silk. So if Maya did not get silk, then that means that silk is ... | General multiplication rule example dependent events Probability & combinatorics.mp3 |
So it is five over six. And then the probability that Doug does not get silk, given that Maya did not get silk. So if Maya did not get silk, then that means that silk is still in the mix, but there's only five possibilities left because Maya picked one of them. And four of them are not silk. They're still silk as an op... | General multiplication rule example dependent events Probability & combinatorics.mp3 |
In the last few videos, we saw that if we had n points, each of them have x and y coordinates. So let me draw n of those points. So let's call this point 1. It has a coordinate x1, y1. You have the second point over here that has a coordinate x2, y2. And then we keep putting points up here and eventually we get to the ... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
It has a coordinate x1, y1. You have the second point over here that has a coordinate x2, y2. And then we keep putting points up here and eventually we get to the nth point over here. So we have the nth point that has the coordinates xn, yn. What we saw is that there is a line that we can find. We can find a line that ... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So we have the nth point that has the coordinates xn, yn. What we saw is that there is a line that we can find. We can find a line that minimizes the squared distance. So this line right here, I'll call it y is equal to mx plus b. That there is some line that minimizes the squared distance to the point. So let me just ... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So this line right here, I'll call it y is equal to mx plus b. That there is some line that minimizes the squared distance to the point. So let me just review what those squared distances are. Sometimes it's called the squared error. So this is the error between the line and point 1. So I'll call that error 1. This is ... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
Sometimes it's called the squared error. So this is the error between the line and point 1. So I'll call that error 1. This is the error between the line and point 2. We'll call this error 2. This is the error between the line and point 3. Sorry, and point n. So if you wanted the total error, if you want the total squa... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
This is the error between the line and point 2. We'll call this error 2. This is the error between the line and point 3. Sorry, and point n. So if you wanted the total error, if you want the total squared error, and this is actually how we started off this whole discussion, the total squared error between the points an... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
Sorry, and point n. So if you wanted the total error, if you want the total squared error, and this is actually how we started off this whole discussion, the total squared error between the points and the line, you literally just take the y value at each point. So for example, you would take y1, that's this value right... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
Well, that point in the line is essentially the y value you get when you substitute x1 into this equation. So I'll just substitute x1 into this equation. So minus mx1 plus b. This right here, that is this y value right over here. That is mx1 plus b. I don't want to get my graph too cluttered, so I'll just delete that t... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
This right here, that is this y value right over here. That is mx1 plus b. I don't want to get my graph too cluttered, so I'll just delete that there. That is error 1 right over there. That is error 1, and we want the squared errors between each of the points in the line. So that's the first one. Then you do the same t... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
That is error 1, and we want the squared errors between each of the points in the line. So that's the first one. Then you do the same thing for the second point. So we started our discussion this way, y2 minus mx2 plus b squared all the way, I'll do dot, dot, dot to show that there are a bunch of these that we have to ... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So we started our discussion this way, y2 minus mx2 plus b squared all the way, I'll do dot, dot, dot to show that there are a bunch of these that we have to do until we get to the nth point, all the way to yn minus mxn plus b squared. Now that we actually know how to find these m's and b's, I showed you the formula, i... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So we can calculate it for a certain set of data. Now what I want to do is kind of come up with a more meaningful estimate of how good this line is fitting the data points that we have. To do that, we're going to ask ourselves the question, how much, or we could even say what percentage, what percentage of the variatio... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
Let's think about this. How much of the total variation in y, there's obviously variation in y. This y value is over here, this point's y value is over here. There's clearly a bunch of variation in the y, but how much of that is essentially described by the variation in x or described by the line? Let's think about tha... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
There's clearly a bunch of variation in the y, but how much of that is essentially described by the variation in x or described by the line? Let's think about that. First let's think about what the total variation is. How much of the, we could even say total variation, how much of the total variation in y? Let's just f... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
How much of the, we could even say total variation, how much of the total variation in y? Let's just figure out what the total variation in y is. The total variation, it's really just a tool for measuring, total variation in y, well we care, when we think about variation, and this is even true when we talk about varian... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
We could just say the total variation in y is just going to be the sum of the distances of each of the y's, so you get y1, let me do this in another color, you get y1, this y1 over here, this is y1 over here, you get y1 minus the mean of all the y's, minus the mean of all the y's squared, plus y2, plus y2, minus the me... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So if you visualize it, you can imagine a line that's y is equal to the mean of y, which would look just like that, and what we're measuring over here, this error right over here is the square of this distance right over here, between this point vertically and this line. The second one is going to be this distance, is ... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
This is the total variation y. Makes sense, if you divide this by n, you actually will get the, I should say this is the total variation in y, if you divide this by n, you're going to get what we typically associate as the variance of y, which is kind of the average square distance. Now we have the total square distanc... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So what we want to do is how much of this, how much of the total variation y is described by the variation in x? So maybe we can think of it this way, so our denominator, we want what percentage of the total variation in y? So let me write it this way. Let me call this as the squared error from the average. Let me call... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
Let me call this as the squared error from the average. Let me call this, this is equal to the squared error, maybe I'll call this the squared error from the mean of y. And this is really the total variation in y. So let's put that as the denominator. Let's put that as the denominator, the total variation y, which is t... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So let's put that as the denominator. Let's put that as the denominator, the total variation y, which is the squared error from the mean of the y's. Now we want to know what percentage of this is described by the variation in x. Now what is not described by the variation in x? We want how much is described by the varia... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
Now what is not described by the variation in x? We want how much is described by the variation in x. But what if we want how much of the total error, how much of the total variation is not described by the line over here, is not described by the regression line. How much of the total data is not? Well, we already have... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
How much of the total data is not? Well, we already have a measure for that. We have the squared error of the line. This tells us the square of the distances from each point to our line. So it is exactly this measure. It tells us how much of the total variation is not described by the regression line. So if you want to... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
This tells us the square of the distances from each point to our line. So it is exactly this measure. It tells us how much of the total variation is not described by the regression line. So if you want to know what percentage of the total variation is not described by the regression line, you would just say, this is th... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So if you want to know what percentage of the total variation is not described by the regression line, you would just say, this is the total, it would just be the squared error, the squared error of the line, because this is the total variation not described by the regression line, divided by the total variation. So le... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
Regression, by the regression line. So to answer our question, what percentage is described by the variation, well, the rest of it has to be described by the variation in x. Because our question is, what percentage of the total variation is described by the variation in x? This is the percentage that is not described. ... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
This is the percentage that is not described. So if this number right here, if this number is, I don't know, 30%, if 30% of the variation in y is not described by the line, then the remainder will be described by the line. So we can essentially just subtract this from 1. So if we take 1 minus the squared error between ... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So if we take 1 minus the squared error between our data points and the line, over the squared error between the data points, between the y's and the mean y, we have, we now have a percentage, this actually tells us what percentage of total variation, total variation, is described by the line. Is described, is describe... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
And this number right here, this is called the coefficient of determination. This is called the coefficient of determination. It's just what statisticians have decided to name it. Coefficient, coefficient of determination. Of determination. Determination. And it's also called r squared. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
Coefficient, coefficient of determination. Of determination. Determination. And it's also called r squared. You might have even heard that term when people talk about regression. Now, let's think about it. If the standard, if the squared error of the line, if the squared error is really small, if the squared error is r... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
And it's also called r squared. You might have even heard that term when people talk about regression. Now, let's think about it. If the standard, if the squared error of the line, if the squared error is really small, if the squared error is really small, what does that mean? It means that these errors, it means that ... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
If the standard, if the squared error of the line, if the squared error is really small, if the squared error is really small, what does that mean? It means that these errors, it means that these errors right over here are really small, are really small, which means that the line is a really good fit. Which means that ... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So if the, let me write it over here. If the squared error of the line is small, is small, it tells us that the line is a good fit. Line is a good, it tells us it's a good fit. Now, what would happen over here? Well, if this number is really small, this is going to be a very small fraction over here. One minus a very s... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
Now, what would happen over here? Well, if this number is really small, this is going to be a very small fraction over here. One minus a very small fraction is going to be a pretty large, it's going to be a number close to one. So then, so then we're going to have our r squared will be close, close to one, which tells ... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So then, so then we're going to have our r squared will be close, close to one, which tells us that a lot of the variation in y is described by the variation in x, which makes sense because the line is a good fit. You take the opposite case. If the squared error of the line is huge, if this number over here is huge, if... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
And so if this number is huge, then this number over here is going to be huge. One minus, or it's going to be a percentage close to one, and one minus that is going to be close to zero. And so if this, if the squared error of the line is large, is large, is large, if this is large, this whole thing is going to be close... | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
But he only has enough money to buy at most four packs. Suppose that each pack has probability 0.2 of containing the card Hugo is hoping for. Let the random variable X be the number of packs of cards Hugo buys. Here is the probability distribution for X. So it looks like there is a 0.2 probability that he buys one pack... | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
Here is the probability distribution for X. So it looks like there is a 0.2 probability that he buys one pack, and that makes sense because that first pack, there is a 0.2 probability that it contains his favorite player's card. And if it does, at that point, he'll just stop. He won't buy any more packs. Now what about... | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
He won't buy any more packs. Now what about the probability that he buys two packs? Well, over here, they give it a 0.16, and that makes sense. There is a 0.8 probability that he does not get the card he wants on the first one, and then there's another 0.2 that he gets it on the second one. So 0.8 times 0.2 does indeed... | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
There is a 0.8 probability that he does not get the card he wants on the first one, and then there's another 0.2 that he gets it on the second one. So 0.8 times 0.2 does indeed equal 0.16. But they're not asking us to calculate that. They give it to us. Then the probability that he gets three packs is 0.128, and then t... | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
They give it to us. Then the probability that he gets three packs is 0.128, and then they've left blank the probability that he gets four packs. But this is the entire discrete probability distribution because Hugo has to stop at four. Even if he doesn't get the card he wants at four on the fourth pack, he's just going... | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
Even if he doesn't get the card he wants at four on the fourth pack, he's just going to stop over there. So we could actually figure out this question mark by just realizing that these four probabilities have to add up to one. But let's just first answer the question. Find the indicated probability. What is the probabi... | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
Find the indicated probability. What is the probability that X is greater than or equal to two? What is the probability? Remember, X is the number of packs of cards Hugo buys. I encourage you to pause the video and try to figure it out. So let's look at the scenarios we're talking about. Probability that our discrete r... | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
Remember, X is the number of packs of cards Hugo buys. I encourage you to pause the video and try to figure it out. So let's look at the scenarios we're talking about. Probability that our discrete random variable X is greater than or equal to two. Well, that's these three scenarios right over here. And so what is thei... | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
Probability that our discrete random variable X is greater than or equal to two. Well, that's these three scenarios right over here. And so what is their combined probability? Well, you might want to say, hey, we need to figure out what the probability of getting exactly four packs are. But we have to remember that the... | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
Well, you might want to say, hey, we need to figure out what the probability of getting exactly four packs are. But we have to remember that these all add up to 100%. And so this right over here is 0.2. And so this is 0.2, the other three combined have to add up to 0.8. 0.8 plus 0.2 is one, or 100%. So just like that, ... | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
And so this is 0.2, the other three combined have to add up to 0.8. 0.8 plus 0.2 is one, or 100%. So just like that, we know that this is 0.8. If for kicks, we wanted to figure out this question mark right over here, we could just say that, look, have to add up to one. So we could say the probability of exactly four is... | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
If for kicks, we wanted to figure out this question mark right over here, we could just say that, look, have to add up to one. So we could say the probability of exactly four is going to be equal to one minus 0.2 minus 0.16 minus 0.128. I get one minus 0.2 minus 0.16 minus 0.128 is equal to 0.512, is equal to 0.512. 0.... | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
0.512. You might immediately say, wait, wait, this seems like a very high probability. There's more than a 50% chance that he buys four packs. And you have to remember, he has to stop at four. Even if on the fourth, he doesn't get the card he wants, he still has to stop there. So there's a high probability that that's ... | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
We have a whole video on it on Khan Academy, but it is an average measure of your blood sugar over roughly a three-month period. So that's the explanatory variable, whether or not you're taking the pill, and the response variable is, well, what does it do to your hemoglobin A1c? We constructed a somewhat classic experi... | Matched pairs experiment design Study design AP Statistics Khan Academy.mp3 |
And to ensure that one group or the other, or I guess both of them, don't end up with an imbalance of, in the case of the last video, an imbalance of men or women, we did what we call block design, where we took our 100 people, and we just happened to have 60 women and 40 men, and we said, okay, well, let's split the 6... | Matched pairs experiment design Study design AP Statistics Khan Academy.mp3 |
Now, this was a pretty good, and it's a bit of a classic experimental design. We would also do it so that the patients don't know which one they're getting, placebo or the actual treatment, so it's a blind experiment. And it would probably be good if even the nurses or the doctors who are administering the pills, who a... | Matched pairs experiment design Study design AP Statistics Khan Academy.mp3 |
But this doesn't mean that it's a perfect experiment, and there seldom is a perfect experiment, and that's why it should be able to be replicated. Other people should try to prove the same thing, maybe in different ways. But even the way that we designed it, there's still a possibility that there are some lurking varia... | Matched pairs experiment design Study design AP Statistics Khan Academy.mp3 |
Maybe, you know, we took care to make sure that our distribution of men and women was roughly even across both of these groups, but maybe by, through that random sampling, we got a disproportionate number of young people in the treatment group, and maybe young people responded better to taking a pill. Maybe it changes ... | Matched pairs experiment design Study design AP Statistics Khan Academy.mp3 |
Vera rents bicycles to tourists. She recorded the height in centimeters of each customer and the frame size in centimeters of the bicycle that customer rented. After plotting her results, Vera noticed that the relationship between the two variables was fairly linear, so she used the data to calculate the following leas... | Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
So before I even look at this question, let's just think about what she did. So she had a bunch of customers, and she recorded, given the height of the customer, what size frame that person rented, and so she might have had something like this, where in the horizontal axis, you have height measured in centimeters, and ... | Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
Maybe there was another person of 100 centimeters in height who got a frame that was slightly larger, and she plotted it there, and then she did a least squares regression, and a least squares regression is trying to fit a line to this data. Oftentimes, you would use a spreadsheet or you use a computer, and that line i... | Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
It might look something like, it might look something like this. So let me plot it. So this, that would be the line, so our regression line, y-hat, is equal to 1 3rd plus 1 3rd x, and so you could view this as a way of predicting or either modeling the relationship or predicting that, hey, if I get a new person, I coul... | Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
So how do we think about this? Well, the residual is going to be the difference between what they actually produce and what the line, what our regression line would have predicted, so we could say residual, let me write it this way, residual is going to be actual, actual minus predicted. So if predicted is larger than ... | Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
If predicted is smaller than actual, this is going to be a positive number. Well, we know the actual, they tell us that, they tell us that they rent, it's a, the 155-centimeter person rents a bike with a 51-centimeter frame, so this is 51 centimeters, but what is the predicted? Well, that's where we can use our regress... | Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
The predicted, I'll do that in orange, the predicted is going to be equal to 1 3rd plus 1 3rd times a person's height. Their height is 155. That's the predicted. Y hat is what our linear regression predicts, our line predicts, so what is this going to be? This is going to be equal to 1 3rd plus 155 over three, which is... | Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
Y hat is what our linear regression predicts, our line predicts, so what is this going to be? This is going to be equal to 1 3rd plus 155 over three, which is equal to 156 over three, which comes out nicely to 52. So the predicted on our line is 52, and so here, so this person is 155, we can plot them right over here, ... | Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
They're coming in slightly below the line. So they're coming in slightly below the line right there, and that distance, which is, and we can see that they are below the line, so that distance is going to be, or in this case, the residual is going to be negative, so this is going to be negative one. And so if we were to... | Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
So what's the probability of the different possible outcomes or the different possible values for this random variable? And we'll plot them to see how that distribution is spread out amongst those possible outcomes. So let's think about all of the different values that you could get when you flip a fair coin three time... | Constructing a probability distribution for random variable Khan Academy.mp3 |
So you could get all heads. Heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads. You could get heads, tails, tails. You could have tails, heads, head. You could have tails, head, tails. | Constructing a probability distribution for random variable Khan Academy.mp3 |
You could get heads, tails, heads. You could get heads, tails, tails. You could have tails, heads, head. You could have tails, head, tails. You could have tails, tails, heads. And then you could have all tails. So when you do the actual experiment, there's eight equally likely outcomes here. | Constructing a probability distribution for random variable Khan Academy.mp3 |
You could have tails, head, tails. You could have tails, tails, heads. And then you could have all tails. So when you do the actual experiment, there's eight equally likely outcomes here. But which of them, how would these relate to the value of this random variable? So let's think about what's the probability. There i... | Constructing a probability distribution for random variable Khan Academy.mp3 |
So when you do the actual experiment, there's eight equally likely outcomes here. But which of them, how would these relate to the value of this random variable? So let's think about what's the probability. There is a situation where you have zero heads. So we could say, what's the probability that our random variable ... | Constructing a probability distribution for random variable Khan Academy.mp3 |
There is a situation where you have zero heads. So we could say, what's the probability that our random variable X is equal to 0? Well, that's this situation right over here where you have zero heads. It is one out of the eight equally likely outcomes. So that's going to be 1 over 8. What's the probability that our ran... | Constructing a probability distribution for random variable Khan Academy.mp3 |
It is one out of the eight equally likely outcomes. So that's going to be 1 over 8. What's the probability that our random variable capital X is equal to 1? Well, let's see. Which of these outcomes gets us exactly one head? We have this one right over here. We have that one right over there. | Constructing a probability distribution for random variable Khan Academy.mp3 |
Well, let's see. Which of these outcomes gets us exactly one head? We have this one right over here. We have that one right over there. We have this one right over there. And I think that's all of them. So three out of the eight equally likely outcomes get us to one head, which is the same thing as saying that our rand... | Constructing a probability distribution for random variable Khan Academy.mp3 |
We have that one right over there. We have this one right over there. And I think that's all of them. So three out of the eight equally likely outcomes get us to one head, which is the same thing as saying that our random variable equals 1. So this has a 3 8's probability. Now, what's the probability? I think you're ma... | Constructing a probability distribution for random variable Khan Academy.mp3 |
So three out of the eight equally likely outcomes get us to one head, which is the same thing as saying that our random variable equals 1. So this has a 3 8's probability. Now, what's the probability? I think you're maybe getting the hang for it at this point. What's the probability that our random variable X is going ... | Constructing a probability distribution for random variable Khan Academy.mp3 |
I think you're maybe getting the hang for it at this point. What's the probability that our random variable X is going to be equal to 2? Well, for X to be equal to 2, that means we got two heads when we flipped the coin three times. So this outcome meets that constraint. This outcome would get our random variable to be... | Constructing a probability distribution for random variable Khan Academy.mp3 |
So this outcome meets that constraint. This outcome would get our random variable to be equal to 2. And this outcome would make our random variable equal to 2. And this is three out of the eight equally likely outcomes. So this has a 3 8's probability. And then finally, we could say, what is the probability that our ra... | Constructing a probability distribution for random variable Khan Academy.mp3 |
And this is three out of the eight equally likely outcomes. So this has a 3 8's probability. And then finally, we could say, what is the probability that our random variable X is equal to 3? Well, how does our random variable X equal 3? Well, we would have to get three heads when we flip the coin. So there's only one o... | Constructing a probability distribution for random variable Khan Academy.mp3 |
Well, how does our random variable X equal 3? Well, we would have to get three heads when we flip the coin. So there's only one out of the eight equally likely outcomes that meets that constraint. So it's a 1 8 probability. So now we just have to think about how we plot this to really see how it's distributed. So let m... | Constructing a probability distribution for random variable Khan Academy.mp3 |
So it's a 1 8 probability. So now we just have to think about how we plot this to really see how it's distributed. So let me draw over here on the vertical axis. I'll draw this will be the probability. And it's going to be between 0 and 1. You can have a probability larger than 1. So just like this. | Constructing a probability distribution for random variable Khan Academy.mp3 |
I'll draw this will be the probability. And it's going to be between 0 and 1. You can have a probability larger than 1. So just like this. So let's see. If this is 1 right over here. And let's see. | Constructing a probability distribution for random variable Khan Academy.mp3 |
So just like this. So let's see. If this is 1 right over here. And let's see. Everything here, it looks like it's an eighth. So let's put everything in terms of eighths. So that's half. | Constructing a probability distribution for random variable Khan Academy.mp3 |
And let's see. Everything here, it looks like it's an eighth. So let's put everything in terms of eighths. So that's half. This is a fourth. That's a fourth. That's not quite a fourth. | Constructing a probability distribution for random variable Khan Academy.mp3 |
So that's half. This is a fourth. That's a fourth. That's not quite a fourth. This is a fourth right over here. And then we can do it in terms of eighths. So that's a pretty good rough approximation. | Constructing a probability distribution for random variable Khan Academy.mp3 |
That's not quite a fourth. This is a fourth right over here. And then we can do it in terms of eighths. So that's a pretty good rough approximation. And then over here, we could have the outcomes. And so outcomes, I'll say outcomes for, or let's write this so value. So value for X. | Constructing a probability distribution for random variable Khan Academy.mp3 |
So that's a pretty good rough approximation. And then over here, we could have the outcomes. And so outcomes, I'll say outcomes for, or let's write this so value. So value for X. So X could be 0, 1. Actually, let me do those same colors. X could be 0. | Constructing a probability distribution for random variable Khan Academy.mp3 |
So value for X. So X could be 0, 1. Actually, let me do those same colors. X could be 0. X could be 1. X could be 2. X could be equal to 2. | Constructing a probability distribution for random variable Khan Academy.mp3 |
X could be 0. X could be 1. X could be 2. X could be equal to 2. And X could be equal to 3. These are the possible values for X. And now we're just going to plot the probability. | Constructing a probability distribution for random variable Khan Academy.mp3 |
X could be equal to 2. And X could be equal to 3. These are the possible values for X. And now we're just going to plot the probability. The probability that X has a value of 0 is 1 eighth. So I'll make a little bar right over here that goes up to 1 eighth. So actually, let me draw it like this. | Constructing a probability distribution for random variable Khan Academy.mp3 |
And now we're just going to plot the probability. The probability that X has a value of 0 is 1 eighth. So I'll make a little bar right over here that goes up to 1 eighth. So actually, let me draw it like this. So this is 1 eighth right over here. The probability that X equals 1 is 3 eighths. So that's 2 eighths, 3 eigh... | Constructing a probability distribution for random variable Khan Academy.mp3 |
So actually, let me draw it like this. So this is 1 eighth right over here. The probability that X equals 1 is 3 eighths. So that's 2 eighths, 3 eighths. Gets us right over. Let me do that in that purple color. So probability of 1, that's 3 eighths. | Constructing a probability distribution for random variable Khan Academy.mp3 |
So that's 2 eighths, 3 eighths. Gets us right over. Let me do that in that purple color. So probability of 1, that's 3 eighths. That's right over there. That's 3 eighths. So let me draw that bar. | Constructing a probability distribution for random variable Khan Academy.mp3 |
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