Geometric distribution

A geometric random variable counts trials until something finally works. Roll a die until you get a six, or test parts until one fails inspection — in each case the quantity of interest is the trial number on which the first success occurs. The distribution requires the same conditions as the binomial: each trial has two outcomes, the trials are independent, and the probability of success p stays constant. A binomial fixes the number of trials and counts successes; a geometric fixes the number of successes at one and counts trials.

To get the first success on trial k, the first k − 1 trials must all fail and the kth must succeed, giving P(X = k) = (1 − p)^(k−1) · p. Rolling a fair die until the first six, the chance the first six comes on the fourth roll is (5/6)³ · (1/6) ≈ 0.0965. The related cumulative result is often faster: the probability that it takes more than k trials is just P(X > k) = (1 − p)^k, since that event means the first k trials all failed.

The mean of a geometric distribution is 1/p, which matches intuition — if a success happens one time in six, you expect to wait about six trials. The standard deviation is √(1 − p) / p. The shape is always right-skewed and decreasing, so a success on the very first trial is the single most likely outcome. It is also memoryless: after ten failures, your expected additional wait is still 1/p — past failures do not make a success "due."

AP Statistics tests the geometric distribution in the probability and random variables unit, typically asking you to verify the conditions, compute a probability, and find the expected value and standard deviation. The FE Mechanical exam covers it among the common probability distributions, where the task is recognizing which distribution fits a situation. The cue is the same on both exams: if the question asks how many attempts until the first success, it is geometric, not binomial.