Geometric Distribution Calculator

The geometric distribution models how many independent Bernoulli trials are required to observe the first success. Every trial has only two outcomes, conventionally called success and failure, and the probability of success remains constant from one trial to the next. Coin flips, repeated sales calls, component inspections, and attempts to connect to a service can fit this model when each attempt is independent and the success rate does not change. This calculator uses the version in which X counts trials, so its possible values begin at one. The probability that the first success occurs on trial k is (1 - p) raised to k - 1, multiplied by p. The power term represents k - 1 consecutive failures, and the final factor represents success on the selected trial. For example, with p equal to 0.5, reaching the first success on trial three requires two failures followed by one success, giving 0.5 times 0.5 times 0.5, or 0.125. The cumulative probability P(X less than or equal to k) is one minus (1 - p) raised to k. It answers the practical question of whether a first success will occur within a specified number of attempts. The expected trial number is 1 divided by p. The variance is (1 - p) divided by p squared. A lower success probability therefore produces both a longer expected wait and much greater uncertainty around that wait. A distinctive property of the geometric distribution is memorylessness. After any number of failures, the probability distribution for the additional wait is the same as it was initially. This makes sense only when attempts truly remain independent with constant p. If practice improves performance, equipment wears down, customers differ, or sampling occurs without replacement, the geometric assumptions no longer hold and another model may be more appropriate. Be careful about the alternate convention found in some textbooks and software. That version counts failures before the first success and therefore starts at zero. Its mean is (1 - p) divided by p, one less than the trial-counting mean shown here. Always identify which convention a formula uses before comparing results. For a stable sequence of independent attempts, this calculator gives exact and cumulative probabilities together with the distribution's central tendency and spread.