If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Cumulative geometric probability (greater than a value)

AP.STATS:
UNC‑3 (EU)
,
UNC‑3.E (LO)
,
UNC‑3.E.2 (EK)

## Video transcript

- [Instructor] Emelia registers vehicles for the Department of Transportation. Sports utility vehicles, also known as SUVs, make up 12% of the vehicles she registers. Let V be the number of vehicles Emelia registers in a day until she first registers an SUV. Assume the type of each vehicle is independent. Find the probability that Emelia registers more than four, more than four, vehicles before she registers an SUV. So, let's just first think about what this random variable V is. So, it's the number of vehicles Emelia registers in a day, until she registers an SUV. So, for example if the first person who walks in the line or through the door, has an SUV and they're trying to register it, then V would be equal to one. If the first person isn't an SUV, but the second person is, then V would be equal to two, so forth and so on. So, this right over here is a classic geometric random variable, right over here. So, geometric random variable. We have a very clear success metric for each trial. Do we have a SUV or not? Each trial is independent, they tell us that. They are independent. The probability of success in each trial is constant. We have a 12% success for each new person who comes through the line. Now the reason this is not a binomial random variable, is that we do not have a finite number of trials. Here, we're gonna keep performing trials. We're gonna keep serving people in the line, until we get an SUV. And so, what we have over here, when they say find the probability that Emelia registers more than four vehicles before she registers an SUV. This is the probability that V is greater than four. So, I encourage you like always, pause this video and see if you can work through it. And we're gonna assume, she's not just gonna leave her, I guess her desk, or wherever the things are being registered. She's not going to leave the counter until someone shows up registering an SUV. So, we will just keep looking at people, I guess we could say over multiple days, forever. She'll work for an infinite number of years, just for the sake of this problem, until an SUV actually shows up. So, try to figure this out. Alright, I'm assuming you've had a go and some of you might said, well, isn't this going to be equal to the probability that V is equal to five, plus the probability that V is equal to six, plus the probability that V is equal to seven, and it just goes on and on and on forever. And this is actually true. And you say, well, how do I calculate this? I'm just summing up an infinite number of things. Now the key realization here, is that one way to think about the probability that V is greater than four, is this is the same thing as the probability that V is not less than or equal to four, these two things are equivalent. So, what's the probability that V is not less than or equal to four? This might be a slightly easier thing for you to calculate. Once again, pause the video and see if you can figure it out. Well what's the probability that V is not less than or equal to four? Well that's the same thing as the probability of first four customers, or first four, I guess people. First four, I'll say customers, or I'll say first four cars. The customer's cars, not SUVs. So, this one is feeling pretty straightforward. What's the probability that for each customer she goes to, that they're not an SUV? Well that's one minus 12%, or 88%, or 0.88. And if we want to know the probability of the first four cars are not SUVs. Well that's 0.88 to the fourth power. And so that's all we have to calculate. And so let's get our calculator out. And say I'm going to get, oops. I'm going to get 0.88 and I'm going to raise it to the fourth power and I get, and I'm just going to round it to, the nearest, let's see, do they tell me to round it? Okay, I'll just round it to the nearest, I guess 100th, well, I'll just write it as 0.5997. Is approximately equal to, 0.5997. If you wanted to write this as a percentage it would be approximately 59.97%. So a little bit better than half, than a 50% shot, a little less than a two-thirds shot, that she is going to have to see more than four customers until she sees an SUV.
AP® is a registered trademark of the College Board, which has not reviewed this resource.