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The null hypothesis is the, hey, there's no news here. And so that would say that, you know, it's kind of the status quo, that the proportion of people who have internet is still the same as the last study, is still the same at 90%. Or I could write 90%, or I could write 0.9 right over here. Now some of you might have been tempted to put 92% there. But it's very important to realize, 92% is the sample proportion, that's the sample statistic. When we're writing these hypotheses, this is about, these are hypotheses about the true parameter. What is the proportion of, the true proportion of homes in California that now have the internet? | Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3 |
Now some of you might have been tempted to put 92% there. But it's very important to realize, 92% is the sample proportion, that's the sample statistic. When we're writing these hypotheses, this is about, these are hypotheses about the true parameter. What is the proportion of, the true proportion of homes in California that now have the internet? And so this is about the true proportion. And so the alternative here is that it's now greater than 90%, or I could say it's greater than 0.9. I could have written 90% or 0.9 here. | Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3 |
What is the proportion of, the true proportion of homes in California that now have the internet? And so this is about the true proportion. And so the alternative here is that it's now greater than 90%, or I could say it's greater than 0.9. I could have written 90% or 0.9 here. And so they really, in this question, they wrote this to kind of distract you, to make you think, oh, maybe I have to incorporate this 92% somehow. And once again, how will they use these hypotheses? Well, they will take this sample in which they got 92% of homes samples had access to the internet. | Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3 |
I could have written 90% or 0.9 here. And so they really, in this question, they wrote this to kind of distract you, to make you think, oh, maybe I have to incorporate this 92% somehow. And once again, how will they use these hypotheses? Well, they will take this sample in which they got 92% of homes samples had access to the internet. So this right over here is my sample proportion. And then they're gonna figure out, well, what's the probability of getting this sample proportion for this sample size if we were to assume that the null hypothesis is true? If this probability of getting this is below a threshold, it's below alpha, below our significance level, then we'll reject the null hypothesis, which would suggest the alternative. | Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3 |
So for example, one driver drives one hour a day. Two drivers drive two hours a day. One driver drives three hours a day. It looks like there's five drivers that drive seven hours a day. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? And then they give us some choices. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? | Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3 |
It looks like there's five drivers that drive seven hours a day. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? And then they give us some choices. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? So pause the video and see if you can figure out which of these percentiles is the closest estimate to the percentile rank of a driver with a daily driving time of six hours, looking at this data right over here. All right, now let's work through this together. So when you think about percentile, you really wanna think about, so let me write this down. | Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3 |
Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? So pause the video and see if you can figure out which of these percentiles is the closest estimate to the percentile rank of a driver with a daily driving time of six hours, looking at this data right over here. All right, now let's work through this together. So when you think about percentile, you really wanna think about, so let me write this down. When we're talking about percentile, we're really saying the percentage of the data that, and there's actually two ways that you could compute it. One is the percentage of the data that is below the amount in question, amount in question. The other possibility is the percent of the data that is at or below, that is at or below the amount, the amount in question. | Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3 |
So when you think about percentile, you really wanna think about, so let me write this down. When we're talking about percentile, we're really saying the percentage of the data that, and there's actually two ways that you could compute it. One is the percentage of the data that is below the amount in question, amount in question. The other possibility is the percent of the data that is at or below, that is at or below the amount, the amount in question. So if we look at this right over here, let's just figure out how many data points, what percentage of the data points are below six hours per day? So let's see, there are, I'm just gonna count them. One, two, three, four, five, six, seven. | Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3 |
The other possibility is the percent of the data that is at or below, that is at or below the amount, the amount in question. So if we look at this right over here, let's just figure out how many data points, what percentage of the data points are below six hours per day? So let's see, there are, I'm just gonna count them. One, two, three, four, five, six, seven. So seven of the 14 are below six hours. So we could just say seven, if we use this first technique, we would have seven of the 14 are below six hours per day, and so that would get us a number of 50%, that six hours is at the 50th percentile. If we wanna say what percentage is at that number or below, then we would also count this one. | Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3 |
One, two, three, four, five, six, seven. So seven of the 14 are below six hours. So we could just say seven, if we use this first technique, we would have seven of the 14 are below six hours per day, and so that would get us a number of 50%, that six hours is at the 50th percentile. If we wanna say what percentage is at that number or below, then we would also count this one. So we would say eight or eight out of 14, eight out of 14, which is the same thing as four out of seven and if we wanna write that as a decimal, let's see, seven goes into 4.000, we just need to estimate. So seven goes into 45 times 35, we subtract, we get a five, bring down a zero, goes five times, I guess it's just gonna be.5 repeating. So 55.5555%. | Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3 |
If we wanna say what percentage is at that number or below, then we would also count this one. So we would say eight or eight out of 14, eight out of 14, which is the same thing as four out of seven and if we wanna write that as a decimal, let's see, seven goes into 4.000, we just need to estimate. So seven goes into 45 times 35, we subtract, we get a five, bring down a zero, goes five times, I guess it's just gonna be.5 repeating. So 55.5555%. So either of these would actually be a legitimate response to the percentile rank for the driver with a daily driving time of six hours. It depends on whether you include the six hours or not. So you could say either the 50th percentile or the roughly the 55th, well actually the 56th percentile if you wanted to round to the nearest percentile. | Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3 |
So 55.5555%. So either of these would actually be a legitimate response to the percentile rank for the driver with a daily driving time of six hours. It depends on whether you include the six hours or not. So you could say either the 50th percentile or the roughly the 55th, well actually the 56th percentile if you wanted to round to the nearest percentile. Now if you look at these choices here, lucky for us, there's only one choice that's reasonably close to either one of those and that's the 55th percentile. And it looks like the people who wrote this question went with the calculation of percentile where they include the data point in question. So everything at six hours or less, what percentage of the total data is that? | Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3 |
So let's look at the frequency table below. So let's see, that's the frequency table, and let's see, there's three categories of computer time, just like they told us, minimal, moderate, and extreme. This is before they go to bed or at night. And then they have the three categories of how much they're sleeping, five or few hours per night, five to seven hours per night, or seven or more hours. Okay, so that's fair enough, so let's see what they want us to do. So they tell us, suppose there were 17 people in the study who were both in moderate computer users and got five to seven hours of sleep. So moderate, five to seven. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
And then they have the three categories of how much they're sleeping, five or few hours per night, five to seven hours per night, or seven or more hours. Okay, so that's fair enough, so let's see what they want us to do. So they tell us, suppose there were 17 people in the study who were both in moderate computer users and got five to seven hours of sleep. So moderate, five to seven. So this category right over here, there were 17 people in this category over here. And just to mark that, let me, I copied and pasted this chart onto my scratch pad so I can write on it. So this group, they're telling us this group, and my pen is really acting up, I don't understand what's going on. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
So moderate, five to seven. So this category right over here, there were 17 people in this category over here. And just to mark that, let me, I copied and pasted this chart onto my scratch pad so I can write on it. So this group, they're telling us this group, and my pen is really acting up, I don't understand what's going on. This group right over here, there are 17 people. So that group right over there is 17 people. Now what are they asking us? | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
So this group, they're telling us this group, and my pen is really acting up, I don't understand what's going on. This group right over here, there are 17 people. So that group right over there is 17 people. Now what are they asking us? They're saying, so they're saying, how many people in the study were both extreme computer users and got five to seven hours of sleep round to the nearest whole number? So extreme and got five to seven. Get my scratch pad out. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
Now what are they asking us? They're saying, so they're saying, how many people in the study were both extreme computer users and got five to seven hours of sleep round to the nearest whole number? So extreme and got five to seven. Get my scratch pad out. So they're saying, how many people are in, how many people are in this bucket, in this bucket right over here? I think I have to replace my pen tablet or something, I don't know why it's getting all splotchy like this. So how would we think about this? | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
Get my scratch pad out. So they're saying, how many people are in, how many people are in this bucket, in this bucket right over here? I think I have to replace my pen tablet or something, I don't know why it's getting all splotchy like this. So how would we think about this? There's 17 people in this group, how many people are in this group? Well, they tell us that 17 is 34% of the row, of the row total, so I guess you could say 17 is 34.3% of the moderate, of the moderate computer users. Or you could say that 17 is 30% of the people who slept five to seven hours each night. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
So how would we think about this? There's 17 people in this group, how many people are in this group? Well, they tell us that 17 is 34% of the row, of the row total, so I guess you could say 17 is 34.3% of the moderate, of the moderate computer users. Or you could say that 17 is 30% of the people who slept five to seven hours each night. Or you could say 17 is 10%, is 10% of the total, of the total number of people. So let's just go with that. We could figure out the total number of people. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
Or you could say that 17 is 30% of the people who slept five to seven hours each night. Or you could say 17 is 10%, is 10% of the total, of the total number of people. So let's just go with that. We could figure out the total number of people. So 10%, actually let me write it this way. So 10%, 10% of the total, 10% of total is going to be equal to 17, or that the total, just divide both sides by 10%, is equal to 17 divided by 10%, which is the same thing as 17 over 0.1, which of course is equal to 170. So the total is 170, and they tell us that extreme, extreme computer users represent, extreme computer users who sleep five to seven hours per night represent 11.7%, 11.7% of the total. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
We could figure out the total number of people. So 10%, actually let me write it this way. So 10%, 10% of the total, 10% of total is going to be equal to 17, or that the total, just divide both sides by 10%, is equal to 17 divided by 10%, which is the same thing as 17 over 0.1, which of course is equal to 170. So the total is 170, and they tell us that extreme, extreme computer users represent, extreme computer users who sleep five to seven hours per night represent 11.7%, 11.7% of the total. So to answer their question of how many people are extreme computer users who sleep five to seven hours per night, that's 11.7% of 170. So let's go back over here. So we actually have a little calculator tool here. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
So the total is 170, and they tell us that extreme, extreme computer users represent, extreme computer users who sleep five to seven hours per night represent 11.7%, 11.7% of the total. So to answer their question of how many people are extreme computer users who sleep five to seven hours per night, that's 11.7% of 170. So let's go back over here. So we actually have a little calculator tool here. So it's 11.7%, which is 0.117, times 170, times 170 is, and let me make sure that you can see what I'm doing by scrolling over a little bit, times 170 is equal to 19.89. So if we're rounding to the nearest whole, that's going to be 20 people, 20 people. And then they are going to ask us some questions. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
So we actually have a little calculator tool here. So it's 11.7%, which is 0.117, times 170, times 170 is, and let me make sure that you can see what I'm doing by scrolling over a little bit, times 170 is equal to 19.89. So if we're rounding to the nearest whole, that's going to be 20 people, 20 people. And then they are going to ask us some questions. They say, does the table show evidence of an association between being a minimal computer user and getting seven hours of sleep or more? So let's just look at the chart. So an association between being a minimal computer user and getting seven hours of sleep or more. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
And then they are going to ask us some questions. They say, does the table show evidence of an association between being a minimal computer user and getting seven hours of sleep or more? So let's just look at the chart. So an association between being a minimal computer user and getting seven hours of sleep or more. So it looks like minimal computer users, so these are the minimal computer users who get seven or more hours of sleep. And there's a couple of ways to read this. So you could say that 51% of minimal computer users get seven or more hours of sleep. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
So an association between being a minimal computer user and getting seven hours of sleep or more. So it looks like minimal computer users, so these are the minimal computer users who get seven or more hours of sleep. And there's a couple of ways to read this. So you could say that 51% of minimal computer users get seven or more hours of sleep. You could say that of the people who get seven or more hours of sleep, 55% are minimal computer users. And of course, this one just says that minimal computer users who get seven or more hours of sleep represent 18.3% of all of the people who were surveyed. So let's look at the choices. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
So you could say that 51% of minimal computer users get seven or more hours of sleep. You could say that of the people who get seven or more hours of sleep, 55% are minimal computer users. And of course, this one just says that minimal computer users who get seven or more hours of sleep represent 18.3% of all of the people who were surveyed. So let's look at the choices. And when I just, actually, before I even look at the choices, let's see if there's an association. It does look like, if you look at minimal computer users, it looks like a small percentage, only 16% get five or few hours. A higher percentage gets five to seven hours. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
So let's look at the choices. And when I just, actually, before I even look at the choices, let's see if there's an association. It does look like, if you look at minimal computer users, it looks like a small percentage, only 16% get five or few hours. A higher percentage gets five to seven hours. And 51%, the highest percentage, gets seven or more. So it looks like for minimal computer users, it looks like the distribution is definitely weighted towards getting more sleep. And for example, if we look at the extreme computer users, it's the opposite trend. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
A higher percentage gets five to seven hours. And 51%, the highest percentage, gets seven or more. So it looks like for minimal computer users, it looks like the distribution is definitely weighted towards getting more sleep. And for example, if we look at the extreme computer users, it's the opposite trend. 47% have five or few hours per night, 33% five to seven hours, and then only 19% get seven or more. And it looks like the moderate is someplace in between. So just looking at this, just looking at each of these rows, it looks like there's a trend where if you use less, if you use a computer for less time, you're more likely to have more sleep. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
And for example, if we look at the extreme computer users, it's the opposite trend. 47% have five or few hours per night, 33% five to seven hours, and then only 19% get seven or more. And it looks like the moderate is someplace in between. So just looking at this, just looking at each of these rows, it looks like there's a trend where if you use less, if you use a computer for less time, you're more likely to have more sleep. And likewise, if you use a computer more, you're more likely to have less sleep. Another way to think about it, when you look at the people who are getting seven or more hours of sleep, a majority of them, a majority of them, are minimal computer users. And there's three categories. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
So just looking at this, just looking at each of these rows, it looks like there's a trend where if you use less, if you use a computer for less time, you're more likely to have more sleep. And likewise, if you use a computer more, you're more likely to have less sleep. Another way to think about it, when you look at the people who are getting seven or more hours of sleep, a majority of them, a majority of them, are minimal computer users. And there's three categories. So for 55% to be minimal computer users, it really does feel like the minimal computer users are more, they're definitely more, disproportionately representing the people who are getting seven, or disproportionately represented in this category of seven or more hours of sleep. And you see that the extreme computer users in this category they represent only 20% of this category. So it does look like there is an association between minimal computer use and getting seven or more hours of sleep. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
And there's three categories. So for 55% to be minimal computer users, it really does feel like the minimal computer users are more, they're definitely more, disproportionately representing the people who are getting seven, or disproportionately represented in this category of seven or more hours of sleep. And you see that the extreme computer users in this category they represent only 20% of this category. So it does look like there is an association between minimal computer use and getting seven or more hours of sleep. But let's look at the actual choices they give us. Does the table show evidence of an association between a minimal computer user and getting seven hours of sleep or more? So yes, because 35.1% are extreme computer users and 29.1% of people are moderate. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
So it does look like there is an association between minimal computer use and getting seven or more hours of sleep. But let's look at the actual choices they give us. Does the table show evidence of an association between a minimal computer user and getting seven hours of sleep or more? So yes, because 35.1% are extreme computer users and 29.1% of people are moderate. Well, I go with the yes, but this doesn't seem to really back up the claim. This is just giving us some kind of random data about the percentage that are extreme computer users or moderate computer users. No, well, I already explained why I go with yes, that there does seem to be a trend. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
So yes, because 35.1% are extreme computer users and 29.1% of people are moderate. Well, I go with the yes, but this doesn't seem to really back up the claim. This is just giving us some kind of random data about the percentage that are extreme computer users or moderate computer users. No, well, I already explained why I go with yes, that there does seem to be a trend. And I don't even believe what the statement is because the total column percentages are essentially equal. We see that the total column, that the column percentages, the column percentages are not equal for the various, for people who are getting seven hours or more of sleep. So we see that right over here, 55, 25, 20. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
No, well, I already explained why I go with yes, that there does seem to be a trend. And I don't even believe what the statement is because the total column percentages are essentially equal. We see that the total column, that the column percentages, the column percentages are not equal for the various, for people who are getting seven hours or more of sleep. So we see that right over here, 55, 25, 20. So I won't go with that one either. Yes, because 51% of minimal computer users get seven or more hours of sleep and only 33% of all computer users get seven or more hours. Yeah, I mean, that seems pretty good. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
So we see that right over here, 55, 25, 20. So I won't go with that one either. Yes, because 51% of minimal computer users get seven or more hours of sleep and only 33% of all computer users get seven or more hours. Yeah, I mean, that seems pretty good. 51% of minimal computer users get seven or more hours of sleep and only 33% of all computer users get seven or more hours of sleep. So that looks like a pretty good explanation. So I'll check that, but let's just review all of them. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
Yeah, I mean, that seems pretty good. 51% of minimal computer users get seven or more hours of sleep and only 33% of all computer users get seven or more hours of sleep. So that looks like a pretty good explanation. So I'll check that, but let's just review all of them. No, well, I already said, I think it's yes, but because the total percentage of extreme users who get five to seven hours of sleep is the same as the total percentage of moderate computer users who get five to seven hours of sleep. So that doesn't really mean, it's not touching on the point that we're looking for. Yes, because 55% of people who get 75, who get seven or more hours of minimal computer users and only 35% of all people are minimal computer users. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
So I'll check that, but let's just review all of them. No, well, I already said, I think it's yes, but because the total percentage of extreme users who get five to seven hours of sleep is the same as the total percentage of moderate computer users who get five to seven hours of sleep. So that doesn't really mean, it's not touching on the point that we're looking for. Yes, because 55% of people who get 75, who get seven or more hours of minimal computer users and only 35% of all people are minimal computer users. Actually, I'll go with this as well. Oh yeah, this is a multi-select here, so I could select that one as well. So this one, we're looking at, so here we looked at the percentage of minimal computer users who get seven hours of sleep and we saw that percentage is higher than for the whole population. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
Yes, because 55% of people who get 75, who get seven or more hours of minimal computer users and only 35% of all people are minimal computer users. Actually, I'll go with this as well. Oh yeah, this is a multi-select here, so I could select that one as well. So this one, we're looking at, so here we looked at the percentage of minimal computer users who get seven hours of sleep and we saw that percentage is higher than for the whole population. Here we're looking at the people who get seven or more hours of sleep and we're saying, wow, 55% of them, 55% of them are minimal computer users even though only 35% of all the people are minimal computer users. So I would go with both of these. And so let us check our answer and we got it right. | Analyzing trends in categorical data Probability and Statistics Khan Academy.mp3 |
Let me do it in that same shade of green. I've already defined set A here. And in both cases, I've defined these sets with numbers. Instead of having numbers as being the objects in the set, I could have had farm animals there or famous presidents, but numbers hopefully keep things fairly simple. So I'm going to start with set A. And from set A, I'm going to subtract set B. So this is one way of thinking about the difference between set A and set B. | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
Instead of having numbers as being the objects in the set, I could have had farm animals there or famous presidents, but numbers hopefully keep things fairly simple. So I'm going to start with set A. And from set A, I'm going to subtract set B. So this is one way of thinking about the difference between set A and set B. And when I've written it this way, this essentially says, give me the set of all of the objects that are in A with the things that are in B taken out of that set. So let's think about what that means. So what's in set A with the things that are in B taken out? | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
So this is one way of thinking about the difference between set A and set B. And when I've written it this way, this essentially says, give me the set of all of the objects that are in A with the things that are in B taken out of that set. So let's think about what that means. So what's in set A with the things that are in B taken out? Well, that means let's take set A and take out a 17, a 19, or take out the 17s, the 19s, and the 6s. So we're going to be left with, we're going to have the 5, we're going to have the 3. We're not going to have the 17 because we subtracted out set B. | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
So what's in set A with the things that are in B taken out? Well, that means let's take set A and take out a 17, a 19, or take out the 17s, the 19s, and the 6s. So we're going to be left with, we're going to have the 5, we're going to have the 3. We're not going to have the 17 because we subtracted out set B. 17 is in set B, so take out anything that is in set B. So you get the 5, the 3. See, the 12 is not in set B, so we can keep that in there. | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
We're not going to have the 17 because we subtracted out set B. 17 is in set B, so take out anything that is in set B. So you get the 5, the 3. See, the 12 is not in set B, so we can keep that in there. And then the 19 is in set B, so we're going to take out the 19 as well. And so that is, this right over here is, you could view it as set B subtracted from set A. So one way of thinking about it, like we just said, these are all of the elements that are in set A that are not in set B. | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
See, the 12 is not in set B, so we can keep that in there. And then the 19 is in set B, so we're going to take out the 19 as well. And so that is, this right over here is, you could view it as set B subtracted from set A. So one way of thinking about it, like we just said, these are all of the elements that are in set A that are not in set B. Another way you could think about it is, these are all of the elements that are not in set B but also in set A. So let me make it clear. You could view this as B subtracted from A, or you could view this as the relative complement of set B in A. | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
So one way of thinking about it, like we just said, these are all of the elements that are in set A that are not in set B. Another way you could think about it is, these are all of the elements that are not in set B but also in set A. So let me make it clear. You could view this as B subtracted from A, or you could view this as the relative complement of set B in A. And we're going to talk a lot more about complements in the future, but the complement is the things that are not in B. And so this is saying, look, what are all of the things that are not in B? So you could say, what are all of the things not in B but are in A? | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
You could view this as B subtracted from A, or you could view this as the relative complement of set B in A. And we're going to talk a lot more about complements in the future, but the complement is the things that are not in B. And so this is saying, look, what are all of the things that are not in B? So you could say, what are all of the things not in B but are in A? So once again, if you said all of the things that aren't in B, then you're thinking about all of the numbers in the whole universe that aren't 17, 19, or 6. And actually, you could even think broader. You're not even just thinking about numbers. | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
So you could say, what are all of the things not in B but are in A? So once again, if you said all of the things that aren't in B, then you're thinking about all of the numbers in the whole universe that aren't 17, 19, or 6. And actually, you could even think broader. You're not even just thinking about numbers. You could even be the color orange is not in set B, so that would be in the absolute complement of B. I don't see a zebra here in set B, so that would be its complement. But we're saying, what are the things that are not in B but are in A? And that would be the numbers 5, 3, and 12. | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
You're not even just thinking about numbers. You could even be the color orange is not in set B, so that would be in the absolute complement of B. I don't see a zebra here in set B, so that would be its complement. But we're saying, what are the things that are not in B but are in A? And that would be the numbers 5, 3, and 12. Now, when we visualized this as B subtracted from A, you might be saying, hey, wait, look, look. OK, I could imagine you took the 17 out, you took the 19 out, but what about taking the 6 out? Maybe you've taken a 6 out, or in traditional subtraction, maybe we would end up with a negative number or something. | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
And that would be the numbers 5, 3, and 12. Now, when we visualized this as B subtracted from A, you might be saying, hey, wait, look, look. OK, I could imagine you took the 17 out, you took the 19 out, but what about taking the 6 out? Maybe you've taken a 6 out, or in traditional subtraction, maybe we would end up with a negative number or something. And when you subtract a set, if the set you're subtracting from does not have that element, then taking that element out of it doesn't change it. If I start with set A and take a 6, if I take all the 6s out of set A, it doesn't change it. There was no 6 to begin with. | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
Maybe you've taken a 6 out, or in traditional subtraction, maybe we would end up with a negative number or something. And when you subtract a set, if the set you're subtracting from does not have that element, then taking that element out of it doesn't change it. If I start with set A and take a 6, if I take all the 6s out of set A, it doesn't change it. There was no 6 to begin with. I could take all the zebras out of set A. It will not change it. Now, another way to denote the relative complement of set B in A or B subtracted from A is the notation that I'm about to write. | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
There was no 6 to begin with. I could take all the zebras out of set A. It will not change it. Now, another way to denote the relative complement of set B in A or B subtracted from A is the notation that I'm about to write. We could have written it this way, A, and then we would have had this little figure like this that looks eerily like a division sign, but this also means the difference between set A and B where we're talking about, when we write it this way, we're talking about all of the things in set A that are not in set B or the things in set B taken out of set A or the relative complement of B in A. Now, with that out of the way, let's think about things the other way around. What would B slash, I'll just call it a slash right over here, what would B minus A be? | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
Now, another way to denote the relative complement of set B in A or B subtracted from A is the notation that I'm about to write. We could have written it this way, A, and then we would have had this little figure like this that looks eerily like a division sign, but this also means the difference between set A and B where we're talking about, when we write it this way, we're talking about all of the things in set A that are not in set B or the things in set B taken out of set A or the relative complement of B in A. Now, with that out of the way, let's think about things the other way around. What would B slash, I'll just call it a slash right over here, what would B minus A be? What would B minus A be, which we could also write as B minus A. What would this be equal to? Well, just going back, we could view this as all of the things in B with all of the things in A taken out of it or all of the things, the complement of A that happens to be in B. | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
What would B slash, I'll just call it a slash right over here, what would B minus A be? What would B minus A be, which we could also write as B minus A. What would this be equal to? Well, just going back, we could view this as all of the things in B with all of the things in A taken out of it or all of the things, the complement of A that happens to be in B. Let's think of it as the set B with all the things in A taken out of it. We start with set B, we have a 17, but a 17 is in set A so we have to take the 17 out. Then we have a 19, but there's a 19 in set A so we have to take the 19 out. | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
Well, just going back, we could view this as all of the things in B with all of the things in A taken out of it or all of the things, the complement of A that happens to be in B. Let's think of it as the set B with all the things in A taken out of it. We start with set B, we have a 17, but a 17 is in set A so we have to take the 17 out. Then we have a 19, but there's a 19 in set A so we have to take the 19 out. Then we have a 6. We don't have to take a 6 out of B because the 6 is not in set A so we're left with just the 6. This would be just the set with a single element in it, set 6. | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
Then we have a 19, but there's a 19 in set A so we have to take the 19 out. Then we have a 6. We don't have to take a 6 out of B because the 6 is not in set A so we're left with just the 6. This would be just the set with a single element in it, set 6. Now, let me ask another question. What would the relative complement of A in A be? This is the same thing as A minus A. | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
This would be just the set with a single element in it, set 6. Now, let me ask another question. What would the relative complement of A in A be? This is the same thing as A minus A. This is literally saying let's take set A and then take all of the things that are in set A out of it. I start with a 5, but there's a 5 in set A so I have to take the 5 out. There's a 3, but there's a 3 in set A so I have to take a 3 out. | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
This is the same thing as A minus A. This is literally saying let's take set A and then take all of the things that are in set A out of it. I start with a 5, but there's a 5 in set A so I have to take the 5 out. There's a 3, but there's a 3 in set A so I have to take a 3 out. I'm going to take all of these things out. I'm just going to be left with the empty set, often called the null set. Sometimes the notation for that will look like this. | Relative complement or difference between sets Probability and Statistics Khan Academy.mp3 |
Let's say that you know your probability of making a free throw. You know that the probability, the probability, and let's say the probability of scoring a free throw, just because if I say make and miss, they both start with M, it'll get confusing. So let's say the probability of scoring, score, you know, free throw, is equal to, is going to be, let's say 70%. If we want to write it as a percent, or we could write it as 0.7 if we write it as a decimal. And let's say the probability of missing a free throw then, and this is just gonna come straight out of what we just wrote down. So the probability of missing, of missing a free throw is just going to be 100% minus this. You're either gonna make or miss, you're either gonna score or miss. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
If we want to write it as a percent, or we could write it as 0.7 if we write it as a decimal. And let's say the probability of missing a free throw then, and this is just gonna come straight out of what we just wrote down. So the probability of missing, of missing a free throw is just going to be 100% minus this. You're either gonna make or miss, you're either gonna score or miss. I don't wanna use make and miss because they both start with M. So this is going to be a 30% probability, or if we write it as a decimal, 0.3. One minus 0.7. These are the only two possibilities, so they have to add up to 100%, or they have to add up to one. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
You're either gonna make or miss, you're either gonna score or miss. I don't wanna use make and miss because they both start with M. So this is going to be a 30% probability, or if we write it as a decimal, 0.3. One minus 0.7. These are the only two possibilities, so they have to add up to 100%, or they have to add up to one. Now let's say that you are going to take six attempts. And what we're curious about, what we're curious about is the probability of exactly, exactly two scores, two scores in six attempts. In six, in six attempts. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
These are the only two possibilities, so they have to add up to 100%, or they have to add up to one. Now let's say that you are going to take six attempts. And what we're curious about, what we're curious about is the probability of exactly, exactly two scores, two scores in six attempts. In six, in six attempts. So let's think about what that is. And I encourage you, if you get inspired at any point in this video, you should pause it and you should try to work through what we're asking right now. So this is what we wanna figure out. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
In six, in six attempts. So let's think about what that is. And I encourage you, if you get inspired at any point in this video, you should pause it and you should try to work through what we're asking right now. So this is what we wanna figure out. The probability of exactly two scores in six attempts. So let's think about the way, let's think about the particular ways of getting two scores in six attempts, and think about the probability for any one of those particular ways, and then we can think about, well how many ways can we get two scores in six attempts? So for example, you could get, you could make the first two free throws, so it could be score, score, and then you miss the next four. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
So this is what we wanna figure out. The probability of exactly two scores in six attempts. So let's think about the way, let's think about the particular ways of getting two scores in six attempts, and think about the probability for any one of those particular ways, and then we can think about, well how many ways can we get two scores in six attempts? So for example, you could get, you could make the first two free throws, so it could be score, score, and then you miss the next four. So score, score, and then it's miss, miss, miss, and miss. So what's the probability of this exact thing happening, this exact thing? Well you have a 0.7 chance of making, of scoring on the first one. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
So for example, you could get, you could make the first two free throws, so it could be score, score, and then you miss the next four. So score, score, and then it's miss, miss, miss, and miss. So what's the probability of this exact thing happening, this exact thing? Well you have a 0.7 chance of making, of scoring on the first one. Then you have a 0.7 chance of scoring on the second one. And then you have a 0.3 chance of missing the next four. So the probability of this exact circumstance is going to be what I'm writing down. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
Well you have a 0.7 chance of making, of scoring on the first one. Then you have a 0.7 chance of scoring on the second one. And then you have a 0.3 chance of missing the next four. So the probability of this exact circumstance is going to be what I'm writing down. And hopefully you don't get the multiplication symbols confused with the decimals. I'm trying to write them a little bit higher. So times 0.3. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
So the probability of this exact circumstance is going to be what I'm writing down. And hopefully you don't get the multiplication symbols confused with the decimals. I'm trying to write them a little bit higher. So times 0.3. And what is this going to be equal to? Well this is going to be equal to, this is going to be 0.7 squared times 0.3, times 0.3 to the one, two, three, fourth, to the fourth power. Now is this the only way to get two scores in six attempts? | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
So times 0.3. And what is this going to be equal to? Well this is going to be equal to, this is going to be 0.7 squared times 0.3, times 0.3 to the one, two, three, fourth, to the fourth power. Now is this the only way to get two scores in six attempts? No, there's many ways of getting two scores in six attempts. For example, maybe you miss the first one, the first attempt, and then you make the second attempt, you score. Then you miss the third attempt. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
Now is this the only way to get two scores in six attempts? No, there's many ways of getting two scores in six attempts. For example, maybe you miss the first one, the first attempt, and then you make the second attempt, you score. Then you miss the third attempt. And then you, let's just say you make the fourth attempt. And then you miss the next two. So then you miss and you miss. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
Then you miss the third attempt. And then you, let's just say you make the fourth attempt. And then you miss the next two. So then you miss and you miss. This is another way to get two scores in six attempts. And what's the probability of this happening? Well as we'll see, it's going to be exactly this, it's just we're multiplying in different order. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
So then you miss and you miss. This is another way to get two scores in six attempts. And what's the probability of this happening? Well as we'll see, it's going to be exactly this, it's just we're multiplying in different order. This is going to be 0.3 times 0.7. You have a 30% chance of missing the first one, a 70% chance of making the second one. And then times 0.3, 30% chance of missing the third, times a 70% chance of making the fourth, times a 30% chance for each of the next two misses. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
Well as we'll see, it's going to be exactly this, it's just we're multiplying in different order. This is going to be 0.3 times 0.7. You have a 30% chance of missing the first one, a 70% chance of making the second one. And then times 0.3, 30% chance of missing the third, times a 70% chance of making the fourth, times a 30% chance for each of the next two misses. If you wanted this exact circumstance, this exact circumstance, this is once again going to be 0.7, if you just rearrange the order in which you're multiplying, this is going to be 0.7 squared times 0.3 to the fourth power. So for any one of these particular ways to get exactly two scores in six attempts, the probability is going to be this. So the probability of getting exactly two scores in six attempts, well it's going to be any one of these probabilities times the number of ways you can get six scores, times the number of ways you can get two scores in six attempts. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
And then times 0.3, 30% chance of missing the third, times a 70% chance of making the fourth, times a 30% chance for each of the next two misses. If you wanted this exact circumstance, this exact circumstance, this is once again going to be 0.7, if you just rearrange the order in which you're multiplying, this is going to be 0.7 squared times 0.3 to the fourth power. So for any one of these particular ways to get exactly two scores in six attempts, the probability is going to be this. So the probability of getting exactly two scores in six attempts, well it's going to be any one of these probabilities times the number of ways you can get six scores, times the number of ways you can get two scores in six attempts. Well, if you have, out of six attempts, you're choosing two of them to have scores, how many ways are there? Well, as you can imagine, this is a combinatorics problem. So you could write this as, you could write this, and let me see how I could, you're gonna take six attempts, so you could write this as six choose, or we're trying to, if you're picking from six things, your six attempts, and you're picking two of them, or two of them are going to be, need to be made if you want to meet these circumstances. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
So the probability of getting exactly two scores in six attempts, well it's going to be any one of these probabilities times the number of ways you can get six scores, times the number of ways you can get two scores in six attempts. Well, if you have, out of six attempts, you're choosing two of them to have scores, how many ways are there? Well, as you can imagine, this is a combinatorics problem. So you could write this as, you could write this, and let me see how I could, you're gonna take six attempts, so you could write this as six choose, or we're trying to, if you're picking from six things, your six attempts, and you're picking two of them, or two of them are going to be, need to be made if you want to meet these circumstances. This is gonna tell us the number of different ways you can make two scores in six attempts. And of course, we can write this in kind of the binomial coefficient notation. We could write this as six choose two, six choose two, and we could just apply the formula for combinations. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
So you could write this as, you could write this, and let me see how I could, you're gonna take six attempts, so you could write this as six choose, or we're trying to, if you're picking from six things, your six attempts, and you're picking two of them, or two of them are going to be, need to be made if you want to meet these circumstances. This is gonna tell us the number of different ways you can make two scores in six attempts. And of course, we can write this in kind of the binomial coefficient notation. We could write this as six choose two, six choose two, and we could just apply the formula for combinations. And if this looks completely unfamiliar, I encourage you to look up combinations on Khan Academy, and we go into some detail on the reasoning behind this formula. It actually makes a lot of sense. This is going to be equal to six factorial over two factorial, over two factorial times six minus two factorial. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
We could write this as six choose two, six choose two, and we could just apply the formula for combinations. And if this looks completely unfamiliar, I encourage you to look up combinations on Khan Academy, and we go into some detail on the reasoning behind this formula. It actually makes a lot of sense. This is going to be equal to six factorial over two factorial, over two factorial times six minus two factorial. So six minus two, six minus two factorial. I'll do the factorial in green again. Six minus two factorial, and what's this going to be equal to? | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
This is going to be equal to six factorial over two factorial, over two factorial times six minus two factorial. So six minus two, six minus two factorial. I'll do the factorial in green again. Six minus two factorial, and what's this going to be equal to? This is going to be equal to six times five times four times three times two, and I'll just throw in the one there, although it doesn't change the value, over two times one, and six minus two is four, so that's going to be four factorial. So this right over here is four factorial, so times four times three times two times one. Well, that and that is going to cancel. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
Six minus two factorial, and what's this going to be equal to? This is going to be equal to six times five times four times three times two, and I'll just throw in the one there, although it doesn't change the value, over two times one, and six minus two is four, so that's going to be four factorial. So this right over here is four factorial, so times four times three times two times one. Well, that and that is going to cancel. And then, let's see, six divided by two is three. So this is 15. There's 15 different ways that you could get, that you could pick two things out of six, I guess is one way to say it, or there's 15, did I say 16? | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
Well, that and that is going to cancel. And then, let's see, six divided by two is three. So this is 15. There's 15 different ways that you could get, that you could pick two things out of six, I guess is one way to say it, or there's 15, did I say 16? There's 15, the sixes and the fives, there's 15 different ways that you could pick two things out of six, or another way of thinking about it is there's 15 different ways to make two out of six free throws. Now, the probability for each of those is this right over here. So the probability of exactly two scores in six attempts, well, this is where we deserve a little bit of a drumroll, this is going to be six choose two times 0.7, 0.7 squared. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
There's 15 different ways that you could get, that you could pick two things out of six, I guess is one way to say it, or there's 15, did I say 16? There's 15, the sixes and the fives, there's 15 different ways that you could pick two things out of six, or another way of thinking about it is there's 15 different ways to make two out of six free throws. Now, the probability for each of those is this right over here. So the probability of exactly two scores in six attempts, well, this is where we deserve a little bit of a drumroll, this is going to be six choose two times 0.7, 0.7 squared. That's, this is two, two, you're gonna make two, you're gonna make two, and then it's 0.3, 0.3 to the, 0.3 to the fourth power. Notice, these will necessarily add up to six. So if this right over here was a three, then this right over here would be a three, and then this would be six minus three, or three right over here. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
So the probability of exactly two scores in six attempts, well, this is where we deserve a little bit of a drumroll, this is going to be six choose two times 0.7, 0.7 squared. That's, this is two, two, you're gonna make two, you're gonna make two, and then it's 0.3, 0.3 to the, 0.3 to the fourth power. Notice, these will necessarily add up to six. So if this right over here was a three, then this right over here would be a three, and then this would be six minus three, or three right over here. And now, what is this value? Well, it's going to be equal to, it's going to be equal to, we have our 15, three times five, so we have this business right over here, it's going to be 15 times, times, let's see, in yellow, 0.7 times 0.7 is going to be times 0.49. And let's see, three to the fourth power would be 81. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
So if this right over here was a three, then this right over here would be a three, and then this would be six minus three, or three right over here. And now, what is this value? Well, it's going to be equal to, it's going to be equal to, we have our 15, three times five, so we have this business right over here, it's going to be 15 times, times, let's see, in yellow, 0.7 times 0.7 is going to be times 0.49. And let's see, three to the fourth power would be 81. Three to the fourth power would be 81, but I'm multiplying four decimals, each of them have one space to the right of the decimal point, so I'm gonna have, this is gonna, I'm gonna have four spaces to the right of the decimal, so 0.0081. So, there you go, whatever this number is, and actually, I might as well get a calculator out and calculate it. So this is going to be, this is going to be, let me, so it's 15 times 0.49, times 0.0081, 0.0081, and we get 0.59535. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
And let's see, three to the fourth power would be 81. Three to the fourth power would be 81, but I'm multiplying four decimals, each of them have one space to the right of the decimal point, so I'm gonna have, this is gonna, I'm gonna have four spaces to the right of the decimal, so 0.0081. So, there you go, whatever this number is, and actually, I might as well get a calculator out and calculate it. So this is going to be, this is going to be, let me, so it's 15 times 0.49, times 0.0081, 0.0081, and we get 0.59535. So this is going to be equal to, let me write it down, and actually, maybe I'll, well, I wish I had a little bit more real estate right over here, but I'll write it in a very bold color. This is going to be, oh, actually, I'm kind of out of bold colors. I'll write it in a slightly less bold color. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
So this is going to be, this is going to be, let me, so it's 15 times 0.49, times 0.0081, 0.0081, and we get 0.59535. So this is going to be equal to, let me write it down, and actually, maybe I'll, well, I wish I had a little bit more real estate right over here, but I'll write it in a very bold color. This is going to be, oh, actually, I'm kind of out of bold colors. I'll write it in a slightly less bold color. This is going to be equal to 0.05935, if we wanted the exact number, or we could say this is approximately, if we round to the nearest percentage, this is approximately a 6% chance, 6% probability of getting exactly two scores in the six attempts. I didn't say two or more, I just said exactly two scores in the six attempts, and actually, it's a fairly low probability because I have a pretty high free throw percentage. If someone has this high of a free throw percentage, it's actually reasonably unlikely that they're only going to make two scores in the six attempts. | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
And so like all random variables, this is taking particular outcomes and converting them into numbers. And this random variable, it could take on the value X equals zero, one, two, three, four, or five. And what I wanna do is figure out, well, what's the probability that this random variable takes on zero, can be one, can be two, can be three, can be four, can be five. And so to do that, first let's think about how many possible outcomes are there from flipping a fair coin five times. So let's think about this. So let's write possible outcomes, possible outcomes from five flips. From five flips. | Binomial distribution Probability and Statistics Khan Academy.mp3 |
And so to do that, first let's think about how many possible outcomes are there from flipping a fair coin five times. So let's think about this. So let's write possible outcomes, possible outcomes from five flips. From five flips. These aren't the possible outcomes for the random variable, this is literally the number of possible outcomes for flipping a coin five times. For example, one possible outcome could be tails, heads, tails, heads, tails. Another possible outcome could be heads, heads, heads, tails, tails. | Binomial distribution Probability and Statistics Khan Academy.mp3 |
From five flips. These aren't the possible outcomes for the random variable, this is literally the number of possible outcomes for flipping a coin five times. For example, one possible outcome could be tails, heads, tails, heads, tails. Another possible outcome could be heads, heads, heads, tails, tails. That is one of the equally likely outcomes, that's another one of the equally likely outcomes. How many of these are there? Well, for each flip, you have two possibilities. | Binomial distribution Probability and Statistics Khan Academy.mp3 |
Another possible outcome could be heads, heads, heads, tails, tails. That is one of the equally likely outcomes, that's another one of the equally likely outcomes. How many of these are there? Well, for each flip, you have two possibilities. So let's write this down. So let me, let me, so the first flip, the first flip, there's two possibilities, times two for the second flip, times two for the third flip, actually, I'm not gonna use the time notation, you might get confused with the random variable. Two possibilities for the first flip, two possibilities for the second flip, two possibilities for the third flip, two possibilities for the fourth flip, and then two possibilities for the fifth flip. | Binomial distribution Probability and Statistics Khan Academy.mp3 |
Well, for each flip, you have two possibilities. So let's write this down. So let me, let me, so the first flip, the first flip, there's two possibilities, times two for the second flip, times two for the third flip, actually, I'm not gonna use the time notation, you might get confused with the random variable. Two possibilities for the first flip, two possibilities for the second flip, two possibilities for the third flip, two possibilities for the fourth flip, and then two possibilities for the fifth flip. Or two to the fifth equally likely possibilities from flipping a coin five times, which of course is equal to 32. And so this is going to be helpful because for each of the values that the random variable can take on, we just have to think about, well, how many of these equally likely possibilities would result in the random variable taking on that value? And let's just delve into it to see what we're actually talking about. | Binomial distribution Probability and Statistics Khan Academy.mp3 |
Two possibilities for the first flip, two possibilities for the second flip, two possibilities for the third flip, two possibilities for the fourth flip, and then two possibilities for the fifth flip. Or two to the fifth equally likely possibilities from flipping a coin five times, which of course is equal to 32. And so this is going to be helpful because for each of the values that the random variable can take on, we just have to think about, well, how many of these equally likely possibilities would result in the random variable taking on that value? And let's just delve into it to see what we're actually talking about. All right, and I'll do it in this light. Let me do it in, I'll start in blue. All right, so let's think about the probability that our random variable X is equal to one. | Binomial distribution Probability and Statistics Khan Academy.mp3 |
And let's just delve into it to see what we're actually talking about. All right, and I'll do it in this light. Let me do it in, I'll start in blue. All right, so let's think about the probability that our random variable X is equal to one. Well, actually, let me start with zero. The probability that our random variable X is equal to zero. So that would mean that you got no heads out of the five flips. | Binomial distribution Probability and Statistics Khan Academy.mp3 |
All right, so let's think about the probability that our random variable X is equal to one. Well, actually, let me start with zero. The probability that our random variable X is equal to zero. So that would mean that you got no heads out of the five flips. Well, there's only one way, one out of the 32 equally likely possibilities that you get no heads. That's the one where you just get five, where you get five tails. So this is just going to be, this is going to be equal to one out of the 32 equally likely possibilities. | Binomial distribution Probability and Statistics Khan Academy.mp3 |
So that would mean that you got no heads out of the five flips. Well, there's only one way, one out of the 32 equally likely possibilities that you get no heads. That's the one where you just get five, where you get five tails. So this is just going to be, this is going to be equal to one out of the 32 equally likely possibilities. Now, for this case, to kind of think in terms of kind of the binomial coefficients and combinatorics and all of that, it's much easier to just reason through it. But just so we can think in those terms, it'll be more useful as we go into higher values for our random variable. And it's also, this is all a buildup for the binomial distribution, so you get a sense of where the name comes from. | Binomial distribution Probability and Statistics Khan Academy.mp3 |
So this is just going to be, this is going to be equal to one out of the 32 equally likely possibilities. Now, for this case, to kind of think in terms of kind of the binomial coefficients and combinatorics and all of that, it's much easier to just reason through it. But just so we can think in those terms, it'll be more useful as we go into higher values for our random variable. And it's also, this is all a buildup for the binomial distribution, so you get a sense of where the name comes from. Let's write it in those terms. So this one, this one, this one right over here, the one way to think about that in combinatorics is that you had five flips and you're choosing zero of them to be heads. Five flips and you're choosing zero of them to be heads. | Binomial distribution Probability and Statistics Khan Academy.mp3 |
And it's also, this is all a buildup for the binomial distribution, so you get a sense of where the name comes from. Let's write it in those terms. So this one, this one, this one right over here, the one way to think about that in combinatorics is that you had five flips and you're choosing zero of them to be heads. Five flips and you're choosing zero of them to be heads. And let's verify that five choose zero is indeed one. So five choose zero, let me write it right over here, five choose zero is equal to five factorial over, over, over five minus zero factorial, over, actually over zero factorial times five minus zero factorial. Five minus zero factorial. | Binomial distribution Probability and Statistics Khan Academy.mp3 |
Five flips and you're choosing zero of them to be heads. And let's verify that five choose zero is indeed one. So five choose zero, let me write it right over here, five choose zero is equal to five factorial over, over, over five minus zero factorial, over, actually over zero factorial times five minus zero factorial. Five minus zero factorial. Well this is zero factorial is one by definition and so this is going to be five factorial over five factorial which is going to be equal to one. Once again, I like reasoning through it instead of blindly applying a formula, but I just wanted to show you that these, these two ideas are consistent. So let's keep going. | Binomial distribution Probability and Statistics Khan Academy.mp3 |
Five minus zero factorial. Well this is zero factorial is one by definition and so this is going to be five factorial over five factorial which is going to be equal to one. Once again, I like reasoning through it instead of blindly applying a formula, but I just wanted to show you that these, these two ideas are consistent. So let's keep going. And I'm going to do x equals one all the way up to x equals five and if you are inspired, and I encourage you to be inspired, try to fill out the whole thing. What's the probability that x equals one, two, three, four, or five? So let's go to the probability that x equals two, oh sorry, x equals one. | Binomial distribution Probability and Statistics Khan Academy.mp3 |
So let's keep going. And I'm going to do x equals one all the way up to x equals five and if you are inspired, and I encourage you to be inspired, try to fill out the whole thing. What's the probability that x equals one, two, three, four, or five? So let's go to the probability that x equals two, oh sorry, x equals one. So the probability that x equals one is going to be equal to, well how do you get one head? Well it could be the first one could be head and then the rest of them are going to be tails. The second one could be head and then the rest of them are going to be tails. | Binomial distribution Probability and Statistics Khan Academy.mp3 |
So let's go to the probability that x equals two, oh sorry, x equals one. So the probability that x equals one is going to be equal to, well how do you get one head? Well it could be the first one could be head and then the rest of them are going to be tails. The second one could be head and then the rest of them are going to be tails. I could write them all out, but you could see that there's just five different places to have that one head. So five out of the 32 equally likely outcomes involve one head. So let me write that down. | Binomial distribution Probability and Statistics Khan Academy.mp3 |
The second one could be head and then the rest of them are going to be tails. I could write them all out, but you could see that there's just five different places to have that one head. So five out of the 32 equally likely outcomes involve one head. So let me write that down. So this is going to be equal to, this is going to be equal to five out of the 32 equally likely outcomes which of course is the same thing. This is going to be the same thing as saying, look I got five flips and I'm choosing one of them. I'm choosing one of them to be heads. | Binomial distribution Probability and Statistics Khan Academy.mp3 |
So let me write that down. So this is going to be equal to, this is going to be equal to five out of the 32 equally likely outcomes which of course is the same thing. This is going to be the same thing as saying, look I got five flips and I'm choosing one of them. I'm choosing one of them to be heads. So that over 32. And you could verify that five factorial over one factorial times five minus, actually let me just do it just so that you don't have to take my word for it. So five choose one is equal to five factorial over one factorial, which is just one, times five minus four, sorry five minus one factorial, which is equal to five factorial over four factorial, which is just going to be equal to five. | Binomial distribution Probability and Statistics Khan Academy.mp3 |
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