Geometric random variables introduction | Random variables | AP Statistics | Khan Academy

So, I have two different random variables here, and what I want to do is think about what type of random variables they are.

So, this first random variable X is equal to the number of sixes after 12 rolls of a fair die. Well, this looks pretty much like a binomial random variable. In fact, I'm pretty confident it is a binomial random variable. We could just go down the checklist: the outcome of each trial can be a success or failure.

So, trial outcome: success or failure. It's either going to go either way. The result of each trial is independent from the other ones. Whether I get a six on the third trial is independent of whether I got a six on the first or the second trial. So, results… let me write this. Trial… I'll just do a shorthand: trial results independent. That's an important condition.

Let's see, there are a fixed number of trials. Fixed number of trials: in this case, we're going to have 12 trials. And then the last one is we have the same probability on each trial. Same probability of success on each trial. So, yes, indeed this met all the conditions for being a binomial random variable.

And this was all just a little bit of review about things that we have talked about in other videos. But what about this thing in the salmon color, the random variable Y?

So, this says the number of rolls until we get a six on a fair die. So, this one strikes us as a little bit different, but let's see where it is actually different. So, does it mean that the trial outcomes… that there's a clear success or failure for each trial? Well, yeah, we're just going to keep rolling.

So, each time we roll it's a trial, and success is when we get a six; failure is when we don't get a six. So, the outcome of each trial can be classified as either a success or failure. So, it meets—let me put the checks right over here—it meets this first constraint.

Are the results of each trial independent? Well, whether I get a six on the first roll or the second roll or the third roll, or the first fourth roll or the third roll, the probabilities shouldn't be dependent on whether I did or didn't get a six on a previous roll. So, we have the independence, and we also have the same probability of success on each trial.

In every case, it's a one-sixth probability that I get a six, so this stays constant. And I skipped this third condition for a reason: because we clearly don't have a fixed number of trials over here.

We could just roll 50 times until we get a 6. The probability that we'd have to roll 50 times is very low, but we might have to roll 500 times in order to get a 6. In fact, think about what the minimum value of Y is and what the maximum value of Y is.

So, the minimum value that this random variable can take—I'll just call it min Y—is equal to what? Well, it's going to take at least one roll. So, that's the minimum value. But what is the maximum value for Y? I'll let you think about that.

Well, I've assumed you've thought about it if you paused the video. Well, there is no max value. You can't say, "Oh, it's a billion," because there's some probability that it might take a billion and one rolls. It is a very, very, very, very, very small probability, but there is some probability it could take a googol rolls, a googolplex rolls.

So, you can imagine where this is going. So, this type of random variable, where it meets a lot of the constraints of a binomial random variable, each trial has a clear success or failure outcome, the probability of success on each trial is constant, the trial results are independent of each other, but we don't have a fixed number of trials—in fact, it's a situation we're saying how many trials do we need to have until we get success?

Maybe that's a general way of framing this type of random variable: how many trials until success? While the binomial random variable was how many trials, or how many successes, I should say, how many successes in a finite number of trials.

So, if you see this general form and it meets these conditions, you can feel good it's a binomial random variable. But if you're meeting this condition: clear success or failure outcome, independent trials, constant probability, but we're not talking about the successes in a finite number of trials; we're talking about how many trials until success, then this type of random variable is called a geometric random variable.

And we will see why in future videos it is called geometric because the math that involves the probabilities of various outcomes looks a lot like geometric growth or geometric sequences in series that we look at in other types of mathematics.

And in case I forgot to mention, the reason why we call binomial random variables is because when you think about the probabilities of different outcomes, you have these things called binomial coefficients based on combinatorics, and those come out of things like Pascal's triangle when you take a binomial to ever-increasing powers.

So, that's where those words come from. But in the next few videos, the important thing is to recognize the difference between the two, and then we're going to start thinking about how do we deal with geometric random variables.