Geometric Distribution Calculator

This calculator works out probabilities for the geometric distribution, the model for how many independent attempts it takes to get the first success. Whenever you repeat a trial that either succeeds or fails with the same probability each time, such as flipping a coin until it lands heads, rolling a die until you get a six, or making sales calls until one converts, the number of attempts up to and including the first success follows the geometric distribution. It answers questions like: what is the chance the first success comes exactly on the third try, or what is the chance it takes more than five attempts, and on average how many attempts are needed. This tool computes them. You enter the probability of success on a single trial and a trial number of interest, and the calculator returns the probability that the first success occurs exactly on that trial, the cumulative probability that it occurs on or before that trial, the probability it takes longer, and the expected, or mean, number of trials until the first success. The results update as you type. Use it for probability and statistics study, for modelling waiting times and reliability, or to plan how many attempts a task is likely to take. The expected number of trials has a pleasingly simple form, one divided by the success probability, so an event with a one in five chance takes five attempts on average. A defining feature of the geometric distribution is that it is memoryless: no matter how many failures you have already had, the chance of success on the next trial is unchanged, which is why past failures do not bring a success any closer. The calculations are exact for your inputs, rounded for display.

P(X = k) = (1 - p)^(k-1) x p. Mean = 1 / p. The distribution is memoryless: past failures do not change the next trial. Rounded for display.