Cumulative geometric probability (greater than a value) | AP Statistics | Khan Academy

Amelia 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 Amelia registers in a day until she first registers an SUV. Assume that the type of each vehicle is independent.

Find the probability that Amelia registers more than four vehicles before she registers an SUV.

So, let's just first think about what this random variable V is. It's the number of vehicles Amelia registers in a day until she registers an SUV. 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, and so forth and so on.

This right over here is a classic geometric random variable. We have a very clear success metric for each trial: Do we have an 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 rate for each new person who comes through the line.

Now, the reason why this is not a binomial random variable is that we do not have a finite number of trials here. We're going to keep performing trials; we're going to keep serving people in the line until we get an SUV. So, what we have over here when they say find the probability that Amelia registers more than four vehicles before she registers an SUV is that this is the probability that V is greater than four.

I encourage you, like always, to pause this video and see if you can work through it. We’ll assume that she's just not going to leave her, I guess, her desk or whatever the things are being registered; she's not going to leave the counter until someone shows up registering an SUV. So, we'll 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.

Now, I'm assuming you've had a go, and some of you might say, "Well, isn't this going to be equal to the probability that V is equal to 5 plus the probability that V is equal to 6 plus the probability that V is equal to 7?" And it just goes on and on and on forever. This is actually true.

You might wonder, "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 that 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? That's the same thing as the probability of the first four customers, or first four, I guess, people—first four, I'll say, customers or I'll say first four cars—not being SUVs.

So this one is feeling pretty straightforward. What's the probability that for each customer she goes to, they're not an SUV? Well, that's one minus 12 percent, or 88 percent, or 0.88. If we want to know the probability that the first four cars are not SUVs, well, that's 0.88 to the fourth power.

So that's all we have to calculate. Let’s get our calculator out. I'm going to get, whoops, 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, well, hundredth.

I'll just write it as 0.5997. This is equal to or approximately equal to 0.5997. If you wanted to write this as a percentage, it would be approximately fifty-nine point nine seven percent. So, a little bit better than half—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.