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Understanding Geometric Probability Distribution

The word “geometric” might remind you of the triangles and squares learned about back in ninth grade geometry class. However, elsewhere in mathland, “geometric” simply refers to multiplication. Thus, “geometric probability distribution” will involve the multiplication of probabilities. Let’s consider some examples.

Geometric Probability Distribution -magoosh

The sex of successive newborns can be modeled using a geometric probability distribution. Photo by Thomas Sauzedde.

Geometric probability distribution: an illustration

Consider the following scenario: “A newlywed couple plans to have children, and they will keep having children until the first girl is born. What is the probability that there are zero boys before the first girl, one boy before the first girl, two boys before the first girl, and so on?”

Keep in mind that the probability of having a girl is 0.5, and that the sex of each baby is independent of the last. Thus, we should expect that as the couple has more and more children, the likelihood that each successive child is a boy becomes smaller and smaller. (The situation here is analogous to flipping a coin multiple times; because each toss is independent, we expect that with each successive toss, it’s likelier we will see a mixture of some heads and some tails, rather than a long string of only heads.) We can illustrate this with the following geometric distribution:

geometric probability distribution

A geometric distribution showing the probability of having only boys after n trials.

Thus, as the number of children, n, increases, the probability that all the kids are boys decreases. Let’s now consider this from another angle.

Cumulative geometric probability distribution

“Cumulative” means “adding up.” We can illustrate the situation we’ve described so far by asking the question in a slightly different way. Instead of asking “What is the probability the couple has n boys in a row?” (the above chart), let’s ask, “What is the probability that the couple has a girl on their nth try?” Such a question will yield the following cumulative geometric distribution.

cumulative geometric probability distribution

A cumulative geometric distribution showing the probability of having a girl on the nth try.

We can see that as the couple continues having kids, the probability of having a girl on the nth try approaches 1. Now that we have an idea of what geometric probability refers to, let’s consider it in a bit more depth.

Fundamental assumptions of geometric probability distribution

We can now define the geometric distribution a bit more formally. The geometric distribution is a discrete probability distribution, in that it involves a discrete number of trials. As with the binomial distribution, the outcome of any trial is binary, resulting in either success or failure. In the above example, success was defined as “having a girl,” but we can define success in any number of ways. Here are a few more examples of situations that could be modeled with a geometric distribution:

• The probability that a basketball player makes a free throw is p = 0.333. What is the probability the player makes his first free throw after n throws? What is the expected number of throws he will make before sinking his first shot?

• A gambler rolls a six-sided die, and he wins when he rolls a 3. What is the probability that he rolls a 3 after n rolls? What is the expected number of rolls he has to make before seeing a 3?

For answers to these questions, and how to solve, check out my other article on Understanding the Geometric Distribution Formula.

Geometric distributions: a conclusion

From the above examples, we can summarize the geometric probability as follows.

• The geometric distribution involves a discrete number of successive trials.
• Each trial is independent of the last, with only two possible outcomes, designed success and failure.
• The probability of success, p, is the same for each trial.
• The geometric distribution models the probability of having no success after n trials.
• The cumulative geometric distribution models the probability of achieving success after n trials.

For more practice with the geometric distribution, check out our statistics blog and videos!

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