Geometric Distribution Calculator

Find the chance that the first success lands on trial k and the expected number of trials.

Geometric distribution

Probability of success on one trial Trial number k

P(first success on trial k): —

P(first success by trial k): —

P(more than k trials): —

Expected number of trials: —

Variance: — · Standard deviation: —

The geometric distribution counts the trials up to and including the first success, with a constant probability on each trial.

Enter the probability of success on a single trial.
Set the trial number k you want the probability for.
Read the point probability, the cumulative probability and the expected number of trials.

Common questions about the free online geometric distribution calculator.

It models the number of independent trials up to and including the first success, with the same success probability p on every trial.

P of X equals k is 1 minus p to the power k minus 1, multiplied by p, which is k minus 1 failures followed by one success.

The cumulative probability is 1 minus 1 minus p to the power k, which is the chance of at least one success by trial k.

The mean of the geometric distribution is 1 divided by p, so a success probability of 0.2 gives an average of 5 trials.

The variance is 1 minus p divided by p squared, and its square root is the standard deviation shown in the results.

Past failures do not change the future. Given no success yet, the wait for the next success still follows the same distribution.

Yes. If p is 1 the first trial always succeeds, so the mean is 1 trial, the variance is 0 and every other trial number has zero chance.

It is the chance that the first success takes more than k trials, so it is simply 1 minus the cumulative probability.

Probabilities are computed in ordinary double precision arithmetic and shown to six decimal places.

No, the geometric distribution calculator runs entirely in your browser and sends nothing anywhere.

Last updated: October 10, 2026