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Geometric Distribution
Common Core: S.MD.1

Geometric Distribution

Author: Jonathan Osters
Description:

Calculate probability by using the geometric distribution formula.

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Tutorial

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In this tutorial, you're going to learn about the geometric distribution. The geometric distribution is somewhat similar to the binomial distribution. It has a geometric setting. It's a probability distribution with a particular setting.

So a scenario or an experiment can be considered geometric if it fits the following four criteria. First is that every trial of the chance experiment only has two outcomes, success or failure. And again, success is fairly arbitrary, just like it was with the binomial.

Also, you can sort of rig an experiment to make it two outcomes, even if there's not. For instance, on a die, you could say rolling a 1 is a success, and everything else is a failure. So even though there are six outcomes, you can rig it so that one of them counts as a success, or some event counts as a success, and some event counts as failure. The only requirement is that these two be complementary events.

Every trial has to be independent of each other, which means that the result of one trial doesn't affect the probabilities for any other. You also need a fixed probability of success on every trial. We'll call that number P. And then finally, the variable of interest, what we're looking for, is the number of trials needed in order to achieve your first success. So I often liken this to the lottery, because you're only going to play until you win, but you play, play, play, and you lose, lose, lose, all the way up until you win, and then you stop.

So suppose a soda company is running a promotion called Lucky 7 where people can win free bottles of soda by looking under the cap, and they advertise one in seven wins. And what they mean is that 1 out of every 7 bottles have caps that say winner on them. So what's the probability that a person playing will win within his first three trials? We're going to assume that this person also stops once he wins.

So we can look at this in a tree diagram. He can win on the first trial, and he has a 1/7 probability of doing so, or he can lose and then win on his second bottle, or he could lose, and lose, and then win on his third bottle. All of these are within the first three trials. The only thing we don't want to have happen is for him to lose, lose, and lose.

So by looking at this tree diagram, you can see that the probability that he wins on the first trial where x is the trial that he wins on is 1/7. The probability that he wins on the second trial is 6/7 times 1/7 for this branch on the tree diagram, and then the probability that he wins on the third trial is 6/7 time 6/7 seven times 1/7. When you take all these values and add them together, the answer to the problem ends up being the probability that he wins within the first three trials, 1/7 plus this number plus this number.

However, the thing that I'm wanting you to notice is how do these calculations differ? Well, you'll notice every time there's a 1/7 representing the fact that he won, but every subsequent value, there's another 6/7 fraction introduced into the calculation. So here, there wasn't any. Here, there's one, here, there's two, the next time would be three. So what you should notice is we can actually come up with a formula here.

So if the geometric distribution is appropriate, and X is the number of trials until you get a success, then the probability that it takes you exactly k trials to obtain a success, well, probability of success is p. You only do that once. So you fail, fail, fail, fail, fail every time except for the last time, when you succeed. So fail, fail, fail, fail, fail, and success.

So to recap, geometric probability follows the geometric setting, so two outcomes per trial, success and failure, fixed probability of success on each trial, independent trials, but this time, instead of how many successes, we know we're only going to succeed once, and we're interested in how many trials it takes in order to do that. When that's the case, the probability that it requires any particular number of trials can be found by multiplying the probability of failure together by all the trials, for all the trials except one of them, the last one, and multiplying by the probability of success the one time. Typically, these values are found on a calculator, but we showed that it's not horribly difficult to find them on a tree diagram either. So we talked about the geometric distribution in this tutorial. Good luck, and we'll see you next time.

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