Geometric Distribution

In this tutorial, you're going to learn about the geometric distribution. Specifically you will focus on:

1. Geometric Distribution

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1.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.

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.

A scenario or an experiment can be considered geometric if it fits the following four criteria.

1. First is that every trial of the chance experiment only has two outcomes, success or failure. Success is fairly arbitrary, just like with the binomial. You can rig an experiment to make it two outcomes, even if there's not. For instance, on a die, you could say rolling a one is a success and everything else is a failure. 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.
2. 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.
3. You need a fixed probability of success on every trial. Call that number p.
4. The variable of interest, what you're looking for, is the number of trials needed in order to achieve your first success.

You can 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.

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. What they mean is that one out of every seven bottles have caps that say winner on them.

What's the probability that a person playing will win within his first three trials? Assume that this person also stops once he wins. You can look at this in a tree diagram.

He can win on the first trial and he has a one seventh 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 this third bottle. All of these are within the first three trials. The only thing you don't want to have happen is for him to lose, lose, and lose.

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 one seventh. The probability that he wins on the second trial is 6/7 times 1/7 for this branch on the tree diagram. Then the probability that he wins on the third trial is 6/7 times 6/7 times 1/7.

When you take all these values and add them together, the answer to the problem ends up being 0.370. The probability that he wins within the first three trials. 1/7 plus the 2nd lower tree, plus the third lower tree.

How do these calculations differ?

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. You can actually come up with a formula here.

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 probability of success is p. You only do that once.

You fail, fail, fail, fail, fail every time except for the last time, when you succeed. So fail, fail, fail, fail, fail and success.

Geometric probability follows the geometric setting. Two outcomes per trial, success and failure. Fixed probability of success on each trial. Independent trials. This time, instead of how many successes, you know you’re only going to succeed once.

You’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 show that it's not horribly difficult to find them on a tree diagram..

Good luck.