Probability for a geometric random variable | Random variables | AP Statistics | Khan Academy

Jeremiah makes 25% of the three-point shots he attempts, far better than my percentage for warmup. Jeremiah likes to shoot three-point shots until he successfully makes one. All right, this is a telltale sign of geometric random variables.

How many trials do he have to take until he gets a success? Let M be the number of shots it takes Jeremiah to successfully make his first three-point shot.

Okay, so they're defining the random variable here: the number of shots it takes, the number of trials it takes until we get a successful three-point shot. Assume that the results of each shot are independent. All right, the probability that he makes a given shot is not dependent on whether he made or missed the previous shots.

Find the probability that Jeremiah's first successful shot occurs on his third attempt. So, like always, pause this video and see if you could have a go at it.

All right, now let's work through this together. So we want to find the probability that, so M is the number of shots it takes until Jeremiah makes his first successful one. What they're really asking is to find the probability that M is equal to 3, that his first successful shot occurs on his third attempt.

So M is equal to 3. So that the number of shots it takes Jeremiah, not me, to make a successful first shot is 3. So how do we do this?

Well, what's just the probability of that happening? Well, that means he has to miss his first two shots and then make his third shot. So what's the probability of him missing his first shot? Well, if he has a 1/4 chance of making his shots, he has a 3/4 chance of missing his shots. So this will be 3/4.

So he misses the first shot, times he has to miss the second shot, and then he has to make his third shot. So there you have it, that's the probability: miss, miss, make.

So what is this going to be? This is equal to nine over sixty-fourths. So there you have it. If you wanted to have this as a decimal, we could get a calculator out real fast. So this is nine—whoops—nine divided by 64 is equal to zero, roughly 0.14.

Approximately 0.14, or another way to think about it is roughly a fourteen percent chance, or fourteen percent probability that it takes him, that his first successful shot occurs in his third attempt.