The geometric distribution answers a 'how many tries until' question: if you repeat an independent yes/no experiment with the same success probability each time, how likely is the first success to land on a particular trial? This calculator computes exact, at-most, and at-least probabilities along with the distribution's mean, variance, and standard deviation, so you can go from a per-trial probability straight to a usable answer about timing.
How the geometric formula works
The geometric PMF, P(X = k) = (1 − p)^(k − 1) · p, breaks into two pieces that must both happen. (1 − p)^(k − 1) is the probability that every one of the first k − 1 trials fails — a chain of independent failures, each with probability 1 − p. p is the probability the k-th trial then succeeds. Multiplying the two gives the probability of that specific sequence: fail, fail, …, fail, success. Because there's no fixed trial limit like the binomial distribution's n, k can in theory be any positive integer, with the probability shrinking geometrically (hence the name) as k grows.
Inputs and what they mean
Success probability (p) is entered as a decimal between 0 and 1 (20% is 0.2, not 20), and represents the same fixed probability applied to every trial — it must be greater than 0, since a zero success rate would mean the first success never comes. Trial number (k) is the count you're asking about, always a positive whole number starting at 1 (there's no 'trial 0'). The comparison type — exactly, at most, or at least — determines whether you get one exact-probability answer or a cumulative sum; 'at most k' is often the more practical real-world question ('what's the chance we've closed a sale within our first 3 calls') since it captures every outcome up to a deadline rather than one precise trial.
The memoryless property and when it applies
The geometric distribution requires the same four conditions as any Bernoulli process — independent trials, exactly two outcomes per trial, and a constant success probability — but with an unlimited number of attempts instead of a fixed count. Its most distinctive trait is the memoryless property: knowing that the first success hasn't happened yet by trial m tells you nothing about how many more trials it'll take — the odds reset exactly as they were at the start. This makes the geometric distribution a natural fit for modeling wait times to a first event (a first sale, a first defect, a first customer churn) where each attempt is genuinely independent of the last. It differs from the binomial distribution, which counts successes within a fixed number of trials rather than timing the first one, and from the negative binomial distribution, which generalizes the geometric case to the r-th success instead of just the first.