What Is the Geometric Distribution?

The geometric distribution is a discrete probability distribution that counts the number of Bernoulli trials until one success is obtained. A Bernoulli trial is an independent repeatable event with a fixed probability p of success and probability q=1-p of failure, such as flipping a coin. Examples of variables with a geometric distribution include counting the number of times a pair of dice need to be rolled until 7 or 11 is rolled or examining products on an assembly line until a defect is found.

This is called a geometric distribution because its successive terms form a geometric series. The probability of success on the first trial is p, the probability on the second trial is pq, the probability on the third trial is pq², and so on. The generalized probability for the nth term is pq^n-1 which is the probability of n-1 failures in a row times the probability of success on the final trial. The geometric distribution is a specific example of a negative binomial distribution that counts the number of Bernoulli trials until r successes are obtained. Some texts also refer to it as a Pascal distribution, although others use the term more generally for any negative binomial distribution.

The geometric distribution is the only discrete probability distribution with the no-memory property, which states that the probability is unaffected by what has occurred before. This is a consequence of the independence of the Bernoulli trials. If the variable, for example, is the number of times that a roulette wheel needs to be spun to come up black, the number of times the wheel came up red before the counting starts doesn’t affect the distribution.

The average of a geometric distribution is 1/p. So if the probability of a product on the assembly line being defective is .0025, one would expect to examine 400 products, on average, before finding a defect. The variance of a geometric distribution is q/p2.