Recognizing binomial variables | Random variables | AP Statistics | Khan Academy

What we're going to do in this video is get some practice classifying whether a random variable is a binomial variable, and we're going to do it by looking at a few exercises from Khan Academy.

So this is a manager who oversees 11 female employees and 9 male employees. They need to pick three of these employees to go on a business trip. So the manager places all 20 names in a hat and chooses at random. Let X equal the number of female employees chosen.

So they're going to do three trials, and on each of those trials, you could say success is if they pick a female employee. Then the random variable X is the number of females out of those three. Is X a binomial variable? Why or why not?

So pause this video and see if you can work through this on your own.

All right, now let's go through each choice. Choice A says each trial isn't being classified as a success or failure, so X is not a binomial variable. I disagree with this. Each trial is being classified as a success or failure; it's either going to be female or not. Since we're counting the number of female employees, if in a trial we pick a female, that would be a success. So each trial is being classified as a success or failure.

So this one over here isn't true. There is no fixed number of trials, so X is not a binomial variable. There is a fixed number of trials; they're doing three trials, they're picking three names out of a hat.

The trials are not independent, so X is not a binomial variable. So this is interesting. For example, trial 1, what's the probability of success? Well, there are 20 employees, 20 names in the hat, and 11 of the outcomes would be success. So you have 11/20; it's the probability of success.

But in trial two, what's the probability of success given success in, I'll say, trial one (T1)? Well, if you succeeded in trial one, that means that there's now only ten female names in the hat out of 19. And if you don't have success in trial one, then you'll have 11 out of 19. So your probability does change based on previous outcomes, and so the trials are not independent. Thus, X is not a binomial variable.

So this one is true; the trials are not independent. That violates that condition for being a binomial variable. In order to be a binomial variable, all your trials have to be independent of each other. Therefore, we'd rule this last one out because this last one says that X has a binomial distribution or it does meet all the conditions for being a binomial variable.

Let's do another example. Here we have different scenarios, and I have the conditions for a binomial variable written right over here. So once again, pause the video and look at each of these scenarios for random variables. Look at these conditions and think about whether these random variables are binomial or not.

So let's look at the first one. In a game involving a standard deck of 52 playing cards, an individual randomly draws seven cards without replacement. Let Y be equal to the number of aces drawn.

Well, in our introductory video to binomial variables, we talked about how if we're doing without replacement, your probability of getting an ace on a given trial is going to be dependent on whether you got aces in previous trials. Because if you got an ace in a previous trial, well, that ace then, you're going to have fewer aces in the deck.

So the trials in this case are not independent. Not independent trials. Now, on the other hand, if on every trial you looked at whatever card you got and put it back in the deck, then they would be independent trials. The probability of getting an ace on each trial would be the same, but not when you have without replacement. So this is not binomial right over here, because you don't have independent trials.

The second scenario: sixty percent of a certain species of tomato live after transplanting from pot to garden. Eli transplants 16 of these tomato plants. Assume that the plants live independently of each other. So whether one plant lives isn't dependent on whether another plant lives. Let T equal the number of tomato plants that live.

All right, so let's look at the conditions. The outcome of each trial can be classified as either success or failure. So each trial over here is one of the tomato plants, and we have 16 of those trials. Success is if the tomato plant lives, and failure is if it dies. So we have either success or failure.

Each trial is independent of the others. They tell us the plants live independently of each other, so whether or not a neighboring plant lives or dies doesn't affect whether the plant next to it lives or dies. So each trial is independent of the others. There is a fixed number of trials; yes, we have 16 right over there.

The probability P of success on each trial remains constant. Well, yeah, according to at least the scenario, they're saying that we have a 60% chance for each tomato plant, which is each trial. So it meets all of the conditions right over here. So this one is binomial.

Now, let's look at this third scenario. In a game of luck, a turn consists of a player continuing to roll a pair of six-sided die until they roll a double (two of the same face values). Let X equal the number of rolls in one turn.

So you're going to keep rolling until they roll a double. Well, the thing that jumps out at me is that you don't have a fixed number of trials; not a fixed number of trials. You could say each trial, each roll is a trial. Success is getting a double, which has a fixed probability. Whether or not you get a double on each trial is going to be independent of the previous roll, so it meets all the other constraints.

But it does not meet that there's a fixed number of trials. You're going to keep rolling. There's some chance you might have to roll 20 times, or 200 times, or who knows how many times until they roll a double. So this violates the fixed number of trials condition, so since there is not a fixed number of trials, this variable X is not binomial.