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You could get sunfish, sunfish, sunfish. You could get, well, what else the other type of fish that you have, or the trout. You could have sunfish, sunfish, trout. You can have sunfish, trout, sunfish. You could have sunfish, trout, trout. You could have trout, sunfish, sunfish. You could have trout, sunfish, trout. | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
You can have sunfish, trout, sunfish. You could have sunfish, trout, trout. You could have trout, sunfish, sunfish. You could have trout, sunfish, trout. You could have trout, trout, sunfish. or you could have all trout. And you see here that each of these, each time you go, there's two possibilities, and so when, or each time you try to catch a fish, there's two possibilities. | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
You could have trout, sunfish, trout. You could have trout, trout, sunfish. or you could have all trout. And you see here that each of these, each time you go, there's two possibilities, and so when, or each time you try to catch a fish, there's two possibilities. So if you're doing it three times, there's two times two times two possibilities. One, two, three, four, five, six, seven, eight possibilities here. Now out of these eight equally likely possibilities, how many of them involve you catching at least two sunfish? | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
And you see here that each of these, each time you go, there's two possibilities, and so when, or each time you try to catch a fish, there's two possibilities. So if you're doing it three times, there's two times two times two possibilities. One, two, three, four, five, six, seven, eight possibilities here. Now out of these eight equally likely possibilities, how many of them involve you catching at least two sunfish? Well you catch at least two sunfish in this one, in this one, in that one, in this one, and I think that is it. That is, yep, this is only one sunfish, one sunfish, one sunfish, and no sunfish. So in four out of the eight equally likely outcomes, you catch at least two sunfish. | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
Now out of these eight equally likely possibilities, how many of them involve you catching at least two sunfish? Well you catch at least two sunfish in this one, in this one, in that one, in this one, and I think that is it. That is, yep, this is only one sunfish, one sunfish, one sunfish, and no sunfish. So in four out of the eight equally likely outcomes, you catch at least two sunfish. So your probability of catching at least two sunfish, probability of at least two sunfish, sunfish, is equal to 4 8ths or 1 1.5. So let's see, what's the expected value? Let's say Y is the expected profit from bet. | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
So in four out of the eight equally likely outcomes, you catch at least two sunfish. So your probability of catching at least two sunfish, probability of at least two sunfish, sunfish, is equal to 4 8ths or 1 1.5. So let's see, what's the expected value? Let's say Y is the expected profit from bet. So let's let Y equals, another random variable is equal to expected profit from bet two. So the expected value of our random variable Y, you have a 1 1.5 chance that you win. So you have a 1 1.5 chance of getting $50, and then you have the 1 1.5 chance, the rest of the probability. | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
Let's say Y is the expected profit from bet. So let's let Y equals, another random variable is equal to expected profit from bet two. So the expected value of our random variable Y, you have a 1 1.5 chance that you win. So you have a 1 1.5 chance of getting $50, and then you have the 1 1.5 chance, the rest of the probability. If there's a 1 1.5 chance you win, there's gonna be a 1 minus 1 1.5, or essentially a 1 1.5 chance that you lose. And so this is going, so you have a 1 1.5 chance of having to pay $25. So let's see what this is. | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
So you have a 1 1.5 chance of getting $50, and then you have the 1 1.5 chance, the rest of the probability. If there's a 1 1.5 chance you win, there's gonna be a 1 minus 1 1.5, or essentially a 1 1.5 chance that you lose. And so this is going, so you have a 1 1.5 chance of having to pay $25. So let's see what this is. This is 1 1.5 times 50 plus 1 1.5 times negative 25. This is going to be 25 minus 12.50, minus 12.50, which is equal to 12.50. So your expected value from bet two is 12.50. | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
So let's see what this is. This is 1 1.5 times 50 plus 1 1.5 times negative 25. This is going to be 25 minus 12.50, minus 12.50, which is equal to 12.50. So your expected value from bet two is 12.50. Your friend says he's willing to take both bets. He's willing to take both bets, a combined total of 50 times. If you want to maximize your expected value, what should you do? | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
So your expected value from bet two is 12.50. Your friend says he's willing to take both bets. He's willing to take both bets, a combined total of 50 times. If you want to maximize your expected value, what should you do? Well, bet number two, actually both of them are good bets, I guess your friend isn't that sophisticated, but bet number two has a higher expected payoff. So I would take bet two all of the time. So I would take bet two all of the time. | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
And he says, well we sell four types of flowers. We sell roses, tulips, sunflowers, and lilies. And you say, what type of pots could I put them in? And he says, well, you could pick any flower, and then you could pick any of our three pots. We have brown pots, we have yellow pots, and we have green pots. So the question that I ask to you is, how many types of, I guess, flower and pots put together can you walk out of this florist store with? For example, you could get a rose in a brown pot. | Counting pot and flower scenarios Statistics and probability 7th grade Khan Academy.mp3 |
And he says, well, you could pick any flower, and then you could pick any of our three pots. We have brown pots, we have yellow pots, and we have green pots. So the question that I ask to you is, how many types of, I guess, flower and pots put together can you walk out of this florist store with? For example, you could get a rose in a brown pot. You could get a rose in a green pot. Or you could get a yellow pot that has a sunflower in it. Or a yellow pot that has a lily in it. | Counting pot and flower scenarios Statistics and probability 7th grade Khan Academy.mp3 |
For example, you could get a rose in a brown pot. You could get a rose in a green pot. Or you could get a yellow pot that has a sunflower in it. Or a yellow pot that has a lily in it. So how many scenarios could you walk out of that store with? And like always, I'll encourage you to pause the video and try to figure it out on your own. So let's think through it. | Counting pot and flower scenarios Statistics and probability 7th grade Khan Academy.mp3 |
Or a yellow pot that has a lily in it. So how many scenarios could you walk out of that store with? And like always, I'll encourage you to pause the video and try to figure it out on your own. So let's think through it. So I'll just write the first letters, just to visualize, or just so I don't have to write down everything. So you could have a brown pot. You could have a yellow pot. | Counting pot and flower scenarios Statistics and probability 7th grade Khan Academy.mp3 |
So let's think through it. So I'll just write the first letters, just to visualize, or just so I don't have to write down everything. So you could have a brown pot. You could have a yellow pot. Or you could have a green pot. You definitely have to pick a pot, so you're gonna have one of those. And then for each of these three, there's four possible flowers you could have. | Counting pot and flower scenarios Statistics and probability 7th grade Khan Academy.mp3 |
You could have a yellow pot. Or you could have a green pot. You definitely have to pick a pot, so you're gonna have one of those. And then for each of these three, there's four possible flowers you could have. You could have a rose with a brown pot. You could have a rose with a yellow pot. You could have a rose with a green pot. | Counting pot and flower scenarios Statistics and probability 7th grade Khan Academy.mp3 |
And then for each of these three, there's four possible flowers you could have. You could have a rose with a brown pot. You could have a rose with a yellow pot. You could have a rose with a green pot. You could have a Tulip with a brown pot. A Tulip with a yellow pot. A Tulip with a green pot. | Counting pot and flower scenarios Statistics and probability 7th grade Khan Academy.mp3 |
You could have a rose with a green pot. You could have a Tulip with a brown pot. A Tulip with a yellow pot. A Tulip with a green pot. You could have a Sunflower with each of the three pots. And you could have, or you could have a Lily with a brown pot. A Lily with a yellow pot. | Counting pot and flower scenarios Statistics and probability 7th grade Khan Academy.mp3 |
A Tulip with a green pot. You could have a Sunflower with each of the three pots. And you could have, or you could have a Lily with a brown pot. A Lily with a yellow pot. And a Lily with a green pot. So how many scenarios are we talking about? Well, we had three pots, so we have three pots right over here, and we have four possible flowers to put in the pots. | Counting pot and flower scenarios Statistics and probability 7th grade Khan Academy.mp3 |
A Lily with a yellow pot. And a Lily with a green pot. So how many scenarios are we talking about? Well, we had three pots, so we have three pots right over here, and we have four possible flowers to put in the pots. And so we see that we have four possible flowers for each of the three pots, so it's going to be three times four, three times four possibilities, or 12. And you see them right over here. This is brown with red, brown with, or brown with rose, brown with tulip, brown with sunflower, brown with lily, yellow with rose, yellow with tulip, yellow with sunflower, yellow with lily. | Counting pot and flower scenarios Statistics and probability 7th grade Khan Academy.mp3 |
In other videos, we introduce ourselves to the idea of a density curve, which is a summary of a distribution, a distribution of data, and in the future, we'll also look at things like probability density, but what I want to talk about in this video is think about what we can glean from them, the properties, how we can describe density curves and the distributions they represent. And we have four of them right over here, and the first thing I want to think about is if we can approximate what value would be the middle value, or the median for the data set described by these density curves. So just to remind ourselves, if we have a set of numbers and we order them from least to greatest, the median would be the middle value, or the midway between the middle two values. In a case like this, we want to find the value for which half of the values are above that value and half of the values are below. So when you're looking at a density curve, you'd want to look at the area, and you'd want to say, okay, at what value do we have equal area above and below that value? And so for this one, just eyeballing it, this value right over here would be the median. And in general, if you have a symmetric distribution like this, the median will be right along that line of symmetry. | Median, mean and skew from density curves AP Statistics Khan Academy.mp3 |
In a case like this, we want to find the value for which half of the values are above that value and half of the values are below. So when you're looking at a density curve, you'd want to look at the area, and you'd want to say, okay, at what value do we have equal area above and below that value? And so for this one, just eyeballing it, this value right over here would be the median. And in general, if you have a symmetric distribution like this, the median will be right along that line of symmetry. Here we have a slightly more unusual distribution. This would be called a bimodal distribution, but you have two major lumps right over here. But it is symmetric, and that point of symmetry is right over here, and so this value, once again, would be the median. | Median, mean and skew from density curves AP Statistics Khan Academy.mp3 |
And in general, if you have a symmetric distribution like this, the median will be right along that line of symmetry. Here we have a slightly more unusual distribution. This would be called a bimodal distribution, but you have two major lumps right over here. But it is symmetric, and that point of symmetry is right over here, and so this value, once again, would be the median. Another way to think about it is the area to the left of that value is equal to the area to the right of that value, making it the median. What if we're dealing with non-symmetric distributions? Well, we'd want to do the same principle. | Median, mean and skew from density curves AP Statistics Khan Academy.mp3 |
But it is symmetric, and that point of symmetry is right over here, and so this value, once again, would be the median. Another way to think about it is the area to the left of that value is equal to the area to the right of that value, making it the median. What if we're dealing with non-symmetric distributions? Well, we'd want to do the same principle. We'd want to think at what value is the area on the right and the area on the left equal? And once again, this isn't going to be super exact, but I'm going to try to approximate it. You might be tempted to go right at the top of this lump right over here, but if I were to do that, it's pretty clear, even eyeballing it, that the right area right over here is larger than the left area. | Median, mean and skew from density curves AP Statistics Khan Academy.mp3 |
Well, we'd want to do the same principle. We'd want to think at what value is the area on the right and the area on the left equal? And once again, this isn't going to be super exact, but I'm going to try to approximate it. You might be tempted to go right at the top of this lump right over here, but if I were to do that, it's pretty clear, even eyeballing it, that the right area right over here is larger than the left area. So that would not be the median. If I move the median a little bit over to the right, this may be right around here, this looks a lot closer. Once again, I'm approximating it, but it's reasonable to say that the area here looks pretty close to the area right over there, and if that is the case, then this is going to be the median. | Median, mean and skew from density curves AP Statistics Khan Academy.mp3 |
You might be tempted to go right at the top of this lump right over here, but if I were to do that, it's pretty clear, even eyeballing it, that the right area right over here is larger than the left area. So that would not be the median. If I move the median a little bit over to the right, this may be right around here, this looks a lot closer. Once again, I'm approximating it, but it's reasonable to say that the area here looks pretty close to the area right over there, and if that is the case, then this is going to be the median. Similarly, on this one right over here, maybe right over here. And once again, I'm just approximating it, but that seems reasonable, that this area is equal to that one. Even though this is longer, it's much lower. | Median, mean and skew from density curves AP Statistics Khan Academy.mp3 |
Once again, I'm approximating it, but it's reasonable to say that the area here looks pretty close to the area right over there, and if that is the case, then this is going to be the median. Similarly, on this one right over here, maybe right over here. And once again, I'm just approximating it, but that seems reasonable, that this area is equal to that one. Even though this is longer, it's much lower. This part of the curve is much higher, even though it goes on less to the right. So that's the median. For well-behaved continuous distributions like this, it's going to be the value for which the area to the left and the area to the right are equal. | Median, mean and skew from density curves AP Statistics Khan Academy.mp3 |
Even though this is longer, it's much lower. This part of the curve is much higher, even though it goes on less to the right. So that's the median. For well-behaved continuous distributions like this, it's going to be the value for which the area to the left and the area to the right are equal. But what about the mean? Well, the mean is you take each of the possible values and you weight it by their frequencies, you weight it by their frequencies, and you add all of that up. And so for symmetric distributions, your mean and your median are actually going to be the same. | Median, mean and skew from density curves AP Statistics Khan Academy.mp3 |
For well-behaved continuous distributions like this, it's going to be the value for which the area to the left and the area to the right are equal. But what about the mean? Well, the mean is you take each of the possible values and you weight it by their frequencies, you weight it by their frequencies, and you add all of that up. And so for symmetric distributions, your mean and your median are actually going to be the same. So this is going to be your mean as well, this is going to be your mean as well. If you wanna think about it in terms of physics, the mean would be your balancing point, the point at which you would wanna put a little fulcrum and you would wanna balance the distribution. And so you could put a little fulcrum here and you could imagine that this thing would balance. | Median, mean and skew from density curves AP Statistics Khan Academy.mp3 |
And so for symmetric distributions, your mean and your median are actually going to be the same. So this is going to be your mean as well, this is going to be your mean as well. If you wanna think about it in terms of physics, the mean would be your balancing point, the point at which you would wanna put a little fulcrum and you would wanna balance the distribution. And so you could put a little fulcrum here and you could imagine that this thing would balance. This thing would balance. And that all comes out of this idea of the weighted average of all of these possible values. What about for these less symmetric distributions? | Median, mean and skew from density curves AP Statistics Khan Academy.mp3 |
And so you could put a little fulcrum here and you could imagine that this thing would balance. This thing would balance. And that all comes out of this idea of the weighted average of all of these possible values. What about for these less symmetric distributions? Well, let's think about it over here. Where would I have to put the fulcrum? Or what does our intuition say if we wanted to balance this? | Median, mean and skew from density curves AP Statistics Khan Academy.mp3 |
What about for these less symmetric distributions? Well, let's think about it over here. Where would I have to put the fulcrum? Or what does our intuition say if we wanted to balance this? Well, we have equal areas on either side, but when you have this long tail to the right, it's going to pull the mean to the right of the median in this case. And so our balance point is probably going to be something closer to that. And once again, this is me approximating it, but this would roughly be our mean. | Median, mean and skew from density curves AP Statistics Khan Academy.mp3 |
Or what does our intuition say if we wanted to balance this? Well, we have equal areas on either side, but when you have this long tail to the right, it's going to pull the mean to the right of the median in this case. And so our balance point is probably going to be something closer to that. And once again, this is me approximating it, but this would roughly be our mean. It would sit in this case to the right of our median. Let me make it clear. This median is referring to that, the mean is referring to this. | Median, mean and skew from density curves AP Statistics Khan Academy.mp3 |
And once again, this is me approximating it, but this would roughly be our mean. It would sit in this case to the right of our median. Let me make it clear. This median is referring to that, the mean is referring to this. In this case, because I have this long tail to the left, it's likely that I would have to balance it out right over here. So the mean would be this value right over there. And there's actually a term for these non-symmetric distributions where the mean is varying from the median. | Median, mean and skew from density curves AP Statistics Khan Academy.mp3 |
This median is referring to that, the mean is referring to this. In this case, because I have this long tail to the left, it's likely that I would have to balance it out right over here. So the mean would be this value right over there. And there's actually a term for these non-symmetric distributions where the mean is varying from the median. Distributions like this are referred to as being skewed. And this distribution, where you have the mean to the right of the median, where you have this long tail to the right, this is called right skewed. Now, the technical idea of skewness can get quite complicated, but generally speaking, you can spot it out when you have a long tail on one direction. | Median, mean and skew from density curves AP Statistics Khan Academy.mp3 |
And there's actually a term for these non-symmetric distributions where the mean is varying from the median. Distributions like this are referred to as being skewed. And this distribution, where you have the mean to the right of the median, where you have this long tail to the right, this is called right skewed. Now, the technical idea of skewness can get quite complicated, but generally speaking, you can spot it out when you have a long tail on one direction. That's the direction in which it will be skewed. Or if the mean is to that direction of the median. So the mean is to the right of the median. | Median, mean and skew from density curves AP Statistics Khan Academy.mp3 |
Now, the technical idea of skewness can get quite complicated, but generally speaking, you can spot it out when you have a long tail on one direction. That's the direction in which it will be skewed. Or if the mean is to that direction of the median. So the mean is to the right of the median. So generally speaking, that's going to be a right skewed distribution. So the opposite of that, here the mean is to the left of the median, and we have this long tail on the left of our distribution. So generally speaking, we will describe these as left skewed distributions. | Median, mean and skew from density curves AP Statistics Khan Academy.mp3 |
And just to visualize that, in this video, we will actually plot these, and we'll get a sense of this random variable's probability distribution. So let's do that. So let's see, maybe, actually, maybe I'll do it like this, so that we can see the probabilities. And actually, I can, let me erase this business right over here. Whoops, that's not working. Let me, here, that might work. Let me erase this real fast, these little scribbles that I had offscreen, and then we can actually plot the distribution. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
And actually, I can, let me erase this business right over here. Whoops, that's not working. Let me, here, that might work. Let me erase this real fast, these little scribbles that I had offscreen, and then we can actually plot the distribution. All right, so the one axis, I'm gonna put all of the different outcomes. So let me, that looks like a pretty straight line. And then this axis, I'm going to plot the probabilities. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
Let me erase this real fast, these little scribbles that I had offscreen, and then we can actually plot the distribution. All right, so the one axis, I'm gonna put all of the different outcomes. So let me, that looks like a pretty straight line. And then this axis, I'm going to plot the probabilities. And that looks like a pretty straight line. And let's see what the probabilities are. We have, let's see, it's all gonna be in terms of 30 seconds, and we get as high as 10 30 seconds. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
And then this axis, I'm going to plot the probabilities. And that looks like a pretty straight line. And let's see what the probabilities are. We have, let's see, it's all gonna be in terms of 30 seconds, and we get as high as 10 30 seconds. So let's say this right over here is, that right over there is 10 30 seconds. 10 30 seconds. Halfway up there, we have two 5 30 seconds. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
We have, let's see, it's all gonna be in terms of 30 seconds, and we get as high as 10 30 seconds. So let's say this right over here is, that right over there is 10 30 seconds. 10 30 seconds. Halfway up there, we have two 5 30 seconds. So let's see, that looks like about half. That right over there is 5 30 seconds. And then one 30 second would be about this. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
Halfway up there, we have two 5 30 seconds. So let's see, that looks like about half. That right over there is 5 30 seconds. And then one 30 second would be about this. It's one, two, let's see, how to split it up. One, two, three, actually let me do it a little bit. One, two, three, four, five. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
And then one 30 second would be about this. It's one, two, let's see, how to split it up. One, two, three, actually let me do it a little bit. One, two, three, four, five. All right, so let's say this is one 30 second right over here. And our probabilities. So this right over here, probability, so this is what the values of the random variable could take on. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
One, two, three, four, five. All right, so let's say this is one 30 second right over here. And our probabilities. So this right over here, probability, so this is what the values of the random variable could take on. So I'll just make a little histogram here. So x equals zero. And then, and the probability there, and actually, since I want to do a histogram, it will look like this. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
So this right over here, probability, so this is what the values of the random variable could take on. So I'll just make a little histogram here. So x equals zero. And then, and the probability there, and actually, since I want to do a histogram, it will look like this. Actually, let me do it a little bit different. So, put it right here. So x is equal to zero. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
And then, and the probability there, and actually, since I want to do a histogram, it will look like this. Actually, let me do it a little bit different. So, put it right here. So x is equal to zero. Right there, the probability is one 30 second. And I can shade that in. Now, I have the probability that x equals one. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
So x is equal to zero. Right there, the probability is one 30 second. And I can shade that in. Now, I have the probability that x equals one. X equals one is five 30 seconds. So let me draw that. So five 30 seconds. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
Now, I have the probability that x equals one. X equals one is five 30 seconds. So let me draw that. So five 30 seconds. So, put the bar there. So we shade that in. So this right over here is the probability that x is equal to one, that we get one. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
So five 30 seconds. So, put the bar there. So we shade that in. So this right over here is the probability that x is equal to one, that we get one. That one, exactly one out of the five flips result in heads. Now we have the probability x equals two. X equals two is 10 30 seconds. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
So this right over here is the probability that x is equal to one, that we get one. That one, exactly one out of the five flips result in heads. Now we have the probability x equals two. X equals two is 10 30 seconds. So that's going to look like this. That's going to look like this. Alright. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
X equals two is 10 30 seconds. So that's going to look like this. That's going to look like this. Alright. My best attempt at hand drawing it. So, somehow I like the aesthetics of hand drawn things more. Sometimes if you just get a computer to graph it, I don't know, sometimes it loses a little bit of its personality. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
Alright. My best attempt at hand drawing it. So, somehow I like the aesthetics of hand drawn things more. Sometimes if you just get a computer to graph it, I don't know, sometimes it loses a little bit of its personality. Alright. So that right over there is the probability that we get that x, that our random variable x is equal to two. Then we have the probability that x equals three, which is also 10 30 seconds. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
Sometimes if you just get a computer to graph it, I don't know, sometimes it loses a little bit of its personality. Alright. So that right over there is the probability that we get that x, that our random variable x is equal to two. Then we have the probability that x equals three, which is also 10 30 seconds. So, that is also 10 30 seconds. So let me draw that. This is also 10 30 seconds. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
Then we have the probability that x equals three, which is also 10 30 seconds. So, that is also 10 30 seconds. So let me draw that. This is also 10 30 seconds. Shade this in. Dum da dum da dum. Alright. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
This is also 10 30 seconds. Shade this in. Dum da dum da dum. Alright. I find this strangely therapeutic. Alright. So this is the probability that x is equal to three. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
Alright. I find this strangely therapeutic. Alright. So this is the probability that x is equal to three. Now x equals four, that's five 30 seconds. So we go back right over here. That's five 30 seconds. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
So this is the probability that x is equal to three. Now x equals four, that's five 30 seconds. So we go back right over here. That's five 30 seconds. So, shade that one in. So this right over here is x is equal to four. And then finally the probability that x equals five is one 30 second again. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
That's five 30 seconds. So, shade that one in. So this right over here is x is equal to four. And then finally the probability that x equals five is one 30 second again. Same level as this right over here. Shade it in. So this right over here is our random variable x equaling five. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
And then finally the probability that x equals five is one 30 second again. Same level as this right over here. Shade it in. So this right over here is our random variable x equaling five. And so when you visually show this probability distribution, it's important to realize this is a discrete probability distribution. This is a discrete random variable. It can only take on a finite number of values. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
So this right over here is our random variable x equaling five. And so when you visually show this probability distribution, it's important to realize this is a discrete probability distribution. This is a discrete random variable. It can only take on a finite number of values. Actually, I should say it's a finite discrete random variable. You could have something that takes on discrete values, but in theory it could take on an infinite number of discrete values. You could just keep counting higher and higher and higher. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
It can only take on a finite number of values. Actually, I should say it's a finite discrete random variable. You could have something that takes on discrete values, but in theory it could take on an infinite number of discrete values. You could just keep counting higher and higher and higher. But this is discrete in that it's kind of these particular values. It can't take on any value in between. And it's also finite. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
You could just keep counting higher and higher and higher. But this is discrete in that it's kind of these particular values. It can't take on any value in between. And it's also finite. It can only take on x equals zero, x equals one, x equals two, x equals three, x equals four, or x equals five. And you see when you plot its probability distribution, this discrete probability distribution, you have it, you know, it starts at one 30 second, it goes up, and then it comes back down. And it has this symmetry. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
And it's also finite. It can only take on x equals zero, x equals one, x equals two, x equals three, x equals four, or x equals five. And you see when you plot its probability distribution, this discrete probability distribution, you have it, you know, it starts at one 30 second, it goes up, and then it comes back down. And it has this symmetry. And a distribution like this, this right over here, a discrete distribution like this, we call this a binomial distribution. And we'll talk in the future about why it's called a binomial distribution. But a big clue, actually I'll tell you why it's called a binomial distribution, is that these probabilities, you can get them using binomial coefficients, using combinatorics. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
And it has this symmetry. And a distribution like this, this right over here, a discrete distribution like this, we call this a binomial distribution. And we'll talk in the future about why it's called a binomial distribution. But a big clue, actually I'll tell you why it's called a binomial distribution, is that these probabilities, you can get them using binomial coefficients, using combinatorics. In another video we'll talk about, especially when we talk about the binomial theorem, why we even call those things binomial coefficients. But it's really based on taking powers of binomials in algebra. But this is a very, very, very, very important distribution. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
But a big clue, actually I'll tell you why it's called a binomial distribution, is that these probabilities, you can get them using binomial coefficients, using combinatorics. In another video we'll talk about, especially when we talk about the binomial theorem, why we even call those things binomial coefficients. But it's really based on taking powers of binomials in algebra. But this is a very, very, very, very important distribution. It's very important in statistics because for a lot of discrete processes, one might assume that the underlying distribution is a binomial distribution. And when we get further into statistics, we will talk why people do that. Now if you were to, if you have much more than five cases here, if instead of saying that the number of heads for flipping a coin five times, you said x is equal to the number of heads of flipping a coin five million times, then you can imagine you'd have much, much, you know, the bars would get narrower and narrower relative to the whole hump. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
But this is a very, very, very, very important distribution. It's very important in statistics because for a lot of discrete processes, one might assume that the underlying distribution is a binomial distribution. And when we get further into statistics, we will talk why people do that. Now if you were to, if you have much more than five cases here, if instead of saying that the number of heads for flipping a coin five times, you said x is equal to the number of heads of flipping a coin five million times, then you can imagine you'd have much, much, you know, the bars would get narrower and narrower relative to the whole hump. And what it would start to do, it would start to approach something that looks really, something that looks really like a bell curve. Let me do that in a color that you can see better that I haven't used yet. So it would start to look, so if you had more and more of these, if you had more and more of these possibilities, it's going to start approaching what looks like a bell curve. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
Now if you were to, if you have much more than five cases here, if instead of saying that the number of heads for flipping a coin five times, you said x is equal to the number of heads of flipping a coin five million times, then you can imagine you'd have much, much, you know, the bars would get narrower and narrower relative to the whole hump. And what it would start to do, it would start to approach something that looks really, something that looks really like a bell curve. Let me do that in a color that you can see better that I haven't used yet. So it would start to look, so if you had more and more of these, if you had more and more of these possibilities, it's going to start approaching what looks like a bell curve. And you've probably heard the notion of a bell curve. And the bell curve is a normal distribution. So if you, one way to think about it is the normal distribution is a probability density function. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
So it would start to look, so if you had more and more of these, if you had more and more of these possibilities, it's going to start approaching what looks like a bell curve. And you've probably heard the notion of a bell curve. And the bell curve is a normal distribution. So if you, one way to think about it is the normal distribution is a probability density function. It's a continuous case. So the yellow one, that we're approaching a normal distribution, and a normal distribution in kind of the classical sense is going to keep going on and on. Normal distribution. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
So if you, one way to think about it is the normal distribution is a probability density function. It's a continuous case. So the yellow one, that we're approaching a normal distribution, and a normal distribution in kind of the classical sense is going to keep going on and on. Normal distribution. And it's related to the binomial, you know, a lot of times in statistics, people will assume a normal distribution because they say, okay, it's the product of kind of an almost an infinite number of random processes happening. Here we're taking a coin, we're flipping it five times, but if you imagine kind of molecules interacting or humans interacting, you're saying, oh, there's almost an infinite number of interactions, and then that's going to result in a normal distribution, which is very, very important in science and statistics. Binomial distribution is the discrete version of that. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
Normal distribution. And it's related to the binomial, you know, a lot of times in statistics, people will assume a normal distribution because they say, okay, it's the product of kind of an almost an infinite number of random processes happening. Here we're taking a coin, we're flipping it five times, but if you imagine kind of molecules interacting or humans interacting, you're saying, oh, there's almost an infinite number of interactions, and then that's going to result in a normal distribution, which is very, very important in science and statistics. Binomial distribution is the discrete version of that. And one little point of notion, you know, these are where the distributions are, this is where they come from, this is how they're related. If you kind of think about, as you get more and more trials, the binomial distribution is going to really approach the normal distribution. But it's really important to think about where these things come from, and we'll talk about it much more in the statistics, because it is reasonable to assume an underlying binomial distribution or a normal distribution for a lot of different types of processes, but sometimes it's not. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
Binomial distribution is the discrete version of that. And one little point of notion, you know, these are where the distributions are, this is where they come from, this is how they're related. If you kind of think about, as you get more and more trials, the binomial distribution is going to really approach the normal distribution. But it's really important to think about where these things come from, and we'll talk about it much more in the statistics, because it is reasonable to assume an underlying binomial distribution or a normal distribution for a lot of different types of processes, but sometimes it's not. And, you know, even in things like economics, sometimes people assume a normal distribution when it's actually much more likely that the things on the ends are going to happen, which might lead to things like economic crises or whatever else. But anyway, I don't want to get off topic. The whole point here is just to appreciate, hey, you know, we started with this random variable, the number of heads from flipping a coin five times, and we plotted it and we were able to see, we were able to visualize this binomial distribution, and I'm kind of telling you, I haven't really shown you, that if you were to have many, many more flips and you defined the random variable in a similar way, then this histogram is going to look a lot, this bar chart is going to look a lot more like a bell curve, and if you had essentially an infinite number of them, you would start having a continuous probability distribution, or I should say, probability density function, and that would be, that would get us closer to a normal distribution. | Visualizing a binomial distribution Probability and Statistics Khan Academy.mp3 |
One is when we are dealing with proportions. So I'll write that on the left side right over here. And the other is when we are dealing with means. In the proportion case, when we're doing our significance test, we will set up some null hypothesis that usually deals with the population proportion. We might say it is equal to some value. Let's just call that P sub one. And then maybe you have an alternative hypothesis that well, no, the population proportion is greater than that or less than that or it's just not equal to that. | When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3 |
In the proportion case, when we're doing our significance test, we will set up some null hypothesis that usually deals with the population proportion. We might say it is equal to some value. Let's just call that P sub one. And then maybe you have an alternative hypothesis that well, no, the population proportion is greater than that or less than that or it's just not equal to that. So let me just go with that one. It's not equal to P sub one. And then what we do to actually test, to actually do the significance test is we take a sample from the population. | When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3 |
And then maybe you have an alternative hypothesis that well, no, the population proportion is greater than that or less than that or it's just not equal to that. So let me just go with that one. It's not equal to P sub one. And then what we do to actually test, to actually do the significance test is we take a sample from the population. It's going to have a sample size of N. We need to make sure that we feel good about making the inference. We've talked about the conditions for inference in previous videos multiple times. But from this, we calculate the sample proportion. | When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3 |
And then what we do to actually test, to actually do the significance test is we take a sample from the population. It's going to have a sample size of N. We need to make sure that we feel good about making the inference. We've talked about the conditions for inference in previous videos multiple times. But from this, we calculate the sample proportion. And then from this, we calculate the P value. And the way that we do the P value, remember, the P value is the probability of getting a sample proportion at least this extreme. And if it's below some threshold, we reject the null hypothesis and it suggests the alternative. | When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3 |
But from this, we calculate the sample proportion. And then from this, we calculate the P value. And the way that we do the P value, remember, the P value is the probability of getting a sample proportion at least this extreme. And if it's below some threshold, we reject the null hypothesis and it suggests the alternative. And over here, the way we do that is we find an associated Z value for that P, for that sample proportion. And the way that we calculate it, we say, okay, look, our Z is going to be how many of the sampling distributions, standard deviations, are we away from the mean? And remember, the mean of the sampling distribution is going to be the population proportion. | When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3 |
And if it's below some threshold, we reject the null hypothesis and it suggests the alternative. And over here, the way we do that is we find an associated Z value for that P, for that sample proportion. And the way that we calculate it, we say, okay, look, our Z is going to be how many of the sampling distributions, standard deviations, are we away from the mean? And remember, the mean of the sampling distribution is going to be the population proportion. So here, we got this sample statistic, this sample proportion. The difference between that and the assumed proportion, remember, when we do these significance tests, we try to figure out the probability assuming the null hypothesis is true. And so when we see this P sub zero, this is the assumed proportion from the null hypothesis. | When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3 |
And remember, the mean of the sampling distribution is going to be the population proportion. So here, we got this sample statistic, this sample proportion. The difference between that and the assumed proportion, remember, when we do these significance tests, we try to figure out the probability assuming the null hypothesis is true. And so when we see this P sub zero, this is the assumed proportion from the null hypothesis. So that's the difference between these two, the sample proportion and the assumed proportion. And then you'd wanna divide it by what's often known as the standard error of the statistic, which is just the standard deviation of the sampling distribution of the sample proportion. And this works out well for proportions because in proportions, I can figure out what this is. | When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3 |
And so when we see this P sub zero, this is the assumed proportion from the null hypothesis. So that's the difference between these two, the sample proportion and the assumed proportion. And then you'd wanna divide it by what's often known as the standard error of the statistic, which is just the standard deviation of the sampling distribution of the sample proportion. And this works out well for proportions because in proportions, I can figure out what this is. This is going to be equal to the square root of the assumed population proportion times one minus the assumed population proportion, all of that over N. And then I would use this Z statistic to figure out the P value. And in this case, I would look at both tails of the distribution because I care about how far I am either above or below the assumed population proportion. Now with means, there's definitely some similarities here. | When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3 |
And this works out well for proportions because in proportions, I can figure out what this is. This is going to be equal to the square root of the assumed population proportion times one minus the assumed population proportion, all of that over N. And then I would use this Z statistic to figure out the P value. And in this case, I would look at both tails of the distribution because I care about how far I am either above or below the assumed population proportion. Now with means, there's definitely some similarities here. You will make a null hypothesis. Maybe you assume the population mean is equal to mu one. And then there's going to be an alternative hypothesis that maybe your population mean is not equal to mu one. | When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3 |
Now with means, there's definitely some similarities here. You will make a null hypothesis. Maybe you assume the population mean is equal to mu one. And then there's going to be an alternative hypothesis that maybe your population mean is not equal to mu one. And you're gonna do something very simple. You take your population, take a sample of size N. Instead of calculating a sample proportion, you calculate a sample mean. And actually, you can calculate other things like a sample standard deviation. | When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3 |
And then there's going to be an alternative hypothesis that maybe your population mean is not equal to mu one. And you're gonna do something very simple. You take your population, take a sample of size N. Instead of calculating a sample proportion, you calculate a sample mean. And actually, you can calculate other things like a sample standard deviation. But now you have an issue. You say, well, ideally, I would use a Z statistic. And you could if you were able to say, well, I could take the difference between my sample mean and the assumed mean in the null hypothesis. | When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3 |
And actually, you can calculate other things like a sample standard deviation. But now you have an issue. You say, well, ideally, I would use a Z statistic. And you could if you were able to say, well, I could take the difference between my sample mean and the assumed mean in the null hypothesis. So that would be this right over here. That's what that zero means, the assumed mean from the null hypothesis. And I would then divide by the standard error of the mean, which is another way of saying the standard deviation of the sampling distribution of the sample mean. | When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3 |
And you could if you were able to say, well, I could take the difference between my sample mean and the assumed mean in the null hypothesis. So that would be this right over here. That's what that zero means, the assumed mean from the null hypothesis. And I would then divide by the standard error of the mean, which is another way of saying the standard deviation of the sampling distribution of the sample mean. But this is not so easy to figure out. In order to figure out this, this is going to be the standard deviation of the underlying population divided by the square root of N. We know what N is going to be if we conducted the sample, but we don't know what the standard deviation is. So instead, what we do is we estimate this. | When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3 |
And I would then divide by the standard error of the mean, which is another way of saying the standard deviation of the sampling distribution of the sample mean. But this is not so easy to figure out. In order to figure out this, this is going to be the standard deviation of the underlying population divided by the square root of N. We know what N is going to be if we conducted the sample, but we don't know what the standard deviation is. So instead, what we do is we estimate this. And so we'll take the sample mean. We subtract from that the assumed population mean from the null hypothesis. And we divide by an estimate of this, which is going to be our sample standard deviation divided by the square root of N. But because this is an estimate, we actually get a better result. | When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3 |
Liz's math test included a survey question asking how many hours students spent studying for the test. The scatter plot and trend line below show the relationship between how many hours students spent studying and their score on the test. The line fitted to model the data has a slope of 15. So the line that they're talking about is right here. So this is the scatter plot. This shows that some student who spent some time between half an hour and an hour studying got a little bit less than a 45 on the test. The student here who got a little bit higher than a 60 spent a little under two hours studying. | Interpreting slope of regression line AP Statistics Khan Academy.mp3 |
So the line that they're talking about is right here. So this is the scatter plot. This shows that some student who spent some time between half an hour and an hour studying got a little bit less than a 45 on the test. The student here who got a little bit higher than a 60 spent a little under two hours studying. This student over here who looks like they got like a 94 or a 95 spent over four hours studying. And so then they fit a line to it and this line has a slope of 15. And before I even read these choices, what's the best interpretation of this slope? | Interpreting slope of regression line AP Statistics Khan Academy.mp3 |
The student here who got a little bit higher than a 60 spent a little under two hours studying. This student over here who looks like they got like a 94 or a 95 spent over four hours studying. And so then they fit a line to it and this line has a slope of 15. And before I even read these choices, what's the best interpretation of this slope? Well, if you think this line is indicative of the trend and it does look like that from the scatter plot, that implies that roughly every extra hour that you study is going to improve your score by 15. You could say on average according to this regression. So if we start over here and we were to increase by one hour our score should improve by 15. | Interpreting slope of regression line AP Statistics Khan Academy.mp3 |
And before I even read these choices, what's the best interpretation of this slope? Well, if you think this line is indicative of the trend and it does look like that from the scatter plot, that implies that roughly every extra hour that you study is going to improve your score by 15. You could say on average according to this regression. So if we start over here and we were to increase by one hour our score should improve by 15. And it does indeed look like that. We're going from, we're going in the horizontal direction, we're going one hour, and then in the vertical direction we're going from 45 to 60. So that's how I would interpret it. | Interpreting slope of regression line AP Statistics Khan Academy.mp3 |
So if we start over here and we were to increase by one hour our score should improve by 15. And it does indeed look like that. We're going from, we're going in the horizontal direction, we're going one hour, and then in the vertical direction we're going from 45 to 60. So that's how I would interpret it. 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 scored zero studied for an average of 15 hours. | Interpreting slope of regression line AP Statistics Khan Academy.mp3 |
So that's how I would interpret it. 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 scored zero studied for an average of 15 hours. 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. | Interpreting slope of regression line AP Statistics Khan Academy.mp3 |
The model predicts that the student who scored zero studied for an average of 15 hours. 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 38. So don't like that choice. The model predicts that the score will increase 15 points for each additional hour of study time. | Interpreting slope of regression line AP Statistics Khan Academy.mp3 |
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 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 were thinking about when we were looking at the model. That's what a slope of 15 tells you. You increase studying time by an hour, it increases score by 15 points. | Interpreting slope of regression line AP Statistics Khan Academy.mp3 |
I'm going to start with a fair coin, and I'm going to flip it 4 times. Flip it 4 times. And the first question I want to ask is, what is the probability that I get exactly 1 head? Or heads. Actually, this is one of those confusing things. When you're talking about what side of the coin, even though I've been not doing this consistently, I'm tempted to say if you're saying 1, it feels like you should do the singular, which would be head, but I read up a little bit of it on the internet, and it seems like when you're talking about coins, you really should say 1 heads, which seems a little bit difficult for me, but I'll try to go with that. So what is the probability of getting exactly 1 heads? | Getting exactly two heads (combinatorics) Probability and Statistics Khan Academy.mp3 |
Or heads. Actually, this is one of those confusing things. When you're talking about what side of the coin, even though I've been not doing this consistently, I'm tempted to say if you're saying 1, it feels like you should do the singular, which would be head, but I read up a little bit of it on the internet, and it seems like when you're talking about coins, you really should say 1 heads, which seems a little bit difficult for me, but I'll try to go with that. So what is the probability of getting exactly 1 heads? And I put that in quotes to say, well, we're just talking about 1 head there, but it's called heads when you're dealing with coins. Anyway, I think you get what I'm talking about. Now to think about this, let's think about how many different possible ways we can get 4 flips of a coin. | Getting exactly two heads (combinatorics) Probability and Statistics Khan Academy.mp3 |
So what is the probability of getting exactly 1 heads? And I put that in quotes to say, well, we're just talking about 1 head there, but it's called heads when you're dealing with coins. Anyway, I think you get what I'm talking about. Now to think about this, let's think about how many different possible ways we can get 4 flips of a coin. We're going to have 1 flip, then another flip, then another flip, then another flip. And this first flip has 2 possibilities, it could be heads or tails. The second flip has 2 possibilities. | Getting exactly two heads (combinatorics) Probability and Statistics Khan Academy.mp3 |
Now to think about this, let's think about how many different possible ways we can get 4 flips of a coin. We're going to have 1 flip, then another flip, then another flip, then another flip. And this first flip has 2 possibilities, it could be heads or tails. The second flip has 2 possibilities. It could be heads or tails. The third flip has 2 possibilities, it could be heads or tails. And the fourth flip has 2 possibilities. | Getting exactly two heads (combinatorics) Probability and Statistics Khan Academy.mp3 |
The second flip has 2 possibilities. It could be heads or tails. The third flip has 2 possibilities, it could be heads or tails. And the fourth flip has 2 possibilities. It could be heads or tails. So you have 2 times 2 times 2 times 2, which is equal to 16 possibilities. 16 possible outcomes when you flip a coin 4 times. | Getting exactly two heads (combinatorics) Probability and Statistics Khan Academy.mp3 |
And the fourth flip has 2 possibilities. It could be heads or tails. So you have 2 times 2 times 2 times 2, which is equal to 16 possibilities. 16 possible outcomes when you flip a coin 4 times. 16 possible outcomes. And any one of the possible outcomes would be 1 of 16. So if I wanted to say, so if I were to just say the probability, and I'm just going to not talk about this one head. | Getting exactly two heads (combinatorics) Probability and Statistics Khan Academy.mp3 |
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