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 pqn-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.