Geometric Distribution Calculator

About this calculator

The geometric distribution models the number of independent Bernoulli trials needed to get the first success, where each trial succeeds with probability p. The probability of the first success landing exactly on trial k is P(X=k) = (1−p)^(k−1)·p; the cumulative probability of succeeding within k trials (≤k) is 1−(1−p)^k, and the probability of needing more than k trials follows directly from the survival function (1−p)^k.

The distribution is memoryless — the probability of needing m more trials doesn't depend on how many failures have already occurred — which makes it a natural model for time-to-first-event problems. Quality control engineers use it to model the number of items inspected before finding the first defect; reliability engineers use it for time-to-first-failure in discrete trials; and it appears in sales (calls until first conversion) and in probability courses as the discrete counterpart of the exponential distribution.