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## The geometric distribution

# Probability for a geometric random variable

AP.STATS:

UNC‑3 (EU)

, UNC‑3.E (LO)

, UNC‑3.E.2 (EK)

## Video transcript

- [Instructor] Jeremiah makes 25% of the three-point shots he attempts. Far better than my percentage. For warm up, Jeremiah likes to shoot three-point shots until
he successfully makes one. Alright, this is the telltale signs of a geometric random variables. How many trials do I have to take until I get 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. Alright, 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 can have a go at it. Alright, now let's work
through this together. So, we wanna find the probability. So, M is the number of shots it takes until Jeremiah makes
his first successful one and so, what they're really asking us, find the probability
that M is equal to three, that his first successful shot occurs on his third attempt. So, M is equal to three. So, that the number of
shots it takes Jeremiah to make his first
successful shot is three. So, how do we do this? Well, what's 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 probability 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 and so, what is this going to be? This is equal to nine over 64ths. So, there you have. If you wanted to have this as a decimal, we could get a calculator out real fast, so this is nine divided by 64 is equal to roughly 0.14. Approximately 0.14 or another way to think about it is a roughly 14% chance or 14% probability that his first successful shot occurs in his third attempt.

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