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Geometric Distribution
Common Core: S.MD.1

Geometric Distribution

Author: Ryan Backman

Calculate probability by using the geometric distribution formula.

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Video Transcription

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Hi. This tutorial covers the Geometric Distribution. Geometric distribution is a type of distribution that occurs when the following four conditions are met-- so, one, each trial has only two outcomes; two, the trials are independent; three, the probability of the outcomes does not change; and, four, the variable of interest is the number of trials until the first success. If you've studied the binomial distribution, how are these conditions different from those of a binomial distribution?

Well, they're pretty similar, but, really, there are two differences. The first diff-- the major difference here is number four-- the variable of interest is the number of trials until the first success, OK? If you remember the binomial distribution, the binomial distribution is-- the variable of interest is the number of successes. So, instead of counting successes, we're-- counting the number of successes, now we're counting the number of trials until the first success, OK?

So, once you're done with that first success-- or, once you have that first success-- then, you're done counting, OK? And, really, that affects the first condition. So, for a binomial setting, you need a fixed number of trials, OK? In a geometric distribution, you do not need a fixed number of trials because it could be one trial, but it could be 1,000 trials. OK, because, again, we're just waiting for that first success to come.

All right, so also like a binomial distribution, a geometric distribution can be explained using a formula. And the formula is the probability of k equals 1 minus lowercase p to the power of k minus 1 times p, OK, where k is the number of trials until the first success. So let's say you want to know, well, what's the probability that it will take six trials until you get your first success? In that case, k would equal 6. So it would be the probability of 6, then you put a 6 up here in the exponent. And then, p is-- represents the probability of success.

OK, so let's apply this formula to a situation here. So, after a long day at work, you hop into your car in the parking lot and try to start it up. The engine turns, but it doesn't start. Your battery is dead so you need to jump start your engine. So let's suppose that 73% of people keep jumper cables in their cars. So you go haphazardly from car to car in the parking lot, asking people for jumper cables. OK, so is this situation geometric?

So let's just run through the four conditions. So condition one-- each trial has only two outcomes. Yes, the outcomes-- when you ask somebody, it's either they have jumper cables or they don't, OK? Trials are independent-- so if you ask one person for jumper cables, and they say no, will that influence the next person you ask? Probably not. So I would say that the trials here would be independent.

The probability of the outcomes does not change. So we'll assume that 73% is always going to be your probability of success. And then, your variable of interest is the number of trials until the first success-- yes. Yeah, it-- we're not counting-- we don't need to count the number of successes. We don't need to count how many people have jumper cables. We just want to know, how many times am I going to have to ask until I get the first person that has jumper cables? Because, as soon as that first person has jumper cables, I'm going to try to jump start the car. OK, so, yes, the situation is geometric.

OK, so now, let's calculate a probability. So what is the probability that you will have to ask three people for jumper cables? OK, so if we have to ask three people for jumper cables, we're looking for the probability of 3. OK, so applying the formula, it's going to be 1 minus the probability of success-- the probability of success is 0.73-- to the power of k minus 1-- so, in this case, it would be a 3 minus 1-- so 3 minus 1-- times the probability of success.

OK, so it's 1 minus 0.73 to the power of 3 minus 1 times 0.73. So 1 minus 0.73 is 0.27-- 3 minus 1 is 2 times 0.73, OK? Now this should make sense, based on what you know about probability two is. 0.27 is the probability of failure-- that's the probability that somebody won't have jumper cables. Now, if we're-- if we have to ask three people, that means the first two people will not have jumper cables and the last person will. So basically we're going to have two failures at 27% and then one success at 73%. So-- and we would multiply all those using the special multiplication rule.

All right, so let's do that in the calculator now. So 0.27 to the power of 2 times 0.73. OK, and that ends up being about 0.053-- oK, so approximately equal to 0.053. So there'll be about a 5% chance that I will have to ask exactly three people to get the first pair of jumper cables. OK, and then the last thing that I think is helpful when you're doing it-- when you're dealing with geometric distributions is to figure out the expected number of trials to get the first success.

OK, so remember that E of X stands for the expected value of some variable X. And that-- remember, we could also abbreviate that using the Greek letter mu, which means "the mean." So this is like the average number of people you'll have to ask. And all this is is just 1 over the probability of success, OK? So mu, in this case, for our example, is going to be 1 over 0.73, OK? So let's do that in the calculator. OK, and we end up with about 1.37-- so approximately 1.37. So, on average, we would expect to ask about 1.37 people to get the first pair of jumper cables. OK, you can't ask 1.37 people, but that tells us that, well, I have to ask either one or two, OK? One may be a little more common than two.

All right, so, in this tutorial, we calculated a geometric probability. We also figured out the expected number of trials to get the first success of the mean, or the expected number-- or the expected value of a geometric distribution. So that has been your tutorial on the Geometric Distribution. Thanks for watching.

Terms to Know
Geometric Distribution

A probability distribution of the number of independent trials of a chance experiment it will take until a success. The criteria for a distribution to be geometric are (1) The chance experiment must only have two outcomes (success/failure) per trial, (2) the trials must be independent, (3) there must be a fixed probability of success for each trial, and (4) the variable of interest is the number of trials needed to obtain a "success" for the first time.

Formulas to Know
Geometric Distribution

P left parenthesis X equals k right parenthesis space equals space left parenthesis 1 minus p right parenthesis to the power of k minus 1 end exponent p
k space i s space t h e space n u m b e r space o f space t r i a l s space u n t i l space f i r s t space s u c c e s s
p space i s space t h e space p r o b a b i l i t y space o f space s u c c e s s space