Geometric Distribution Calculator

A geometric experiment is a statistical experiment consisting of repeated independent trials, each with only two possible outcomes: success or failure. The probability of success (p) remains constant on every trial, and the experiment continues until the first success is observed. The random variable X records either the number of the trial on which the first success occurs, or the number of failures before the first success, depending on the convention used.

In the 'number of trials' convention, X counts the total number of trials needed to get the first success, so X ∈ {1, 2, 3, …} and P(X = x) = (1−p)^(x−1) · p. In the 'number of failures' convention, X counts only the failures before the first success, so X ∈ {0, 1, 2, …} and P(X = x) = (1−p)^x · p. Both describe the same underlying process — make sure to choose the convention that matches your textbook or problem statement.

For the trials convention: P(X = x) = (1 − p)^(x − 1) × p, where p is the probability of success on each trial and x ≥ 1. For the failures convention: P(X = x) = (1 − p)^x × p, where x ≥ 0. In both cases, you multiply the probability of getting (x − 1) or x consecutive failures by the probability of success on the final trial.

For the trials convention (X = trial of first success): Mean μ = 1/p and Variance σ² = (1 − p)/p². For the failures convention (X = failures before success): Mean μ = (1 − p)/p and Variance σ² = (1 − p)/p². The standard deviation σ is the square root of the variance in both cases. For example, if p = 0.25, the expected number of trials until the first success is 1/0.25 = 4.

P(X ≤ x) is the cumulative distribution function (CDF). It gives the probability that the first success occurs on or before trial x (trials convention), or that there are at most x failures before the first success (failures convention). For the trials convention, P(X ≤ x) = 1 − (1 − p)^x. This is useful when you want the probability of success within a given number of attempts.

Yes — the geometric distribution is the only discrete probability distribution with the memoryless property. This means that given you have already had k failures, the probability of needing m more trials to get the first success is the same as if you were starting fresh. Formally, P(X > k + m | X > k) = P(X > m). This property makes it analogous to the exponential distribution in continuous probability.

The success probability p must satisfy 0 < p ≤ 1. When p = 1, success is guaranteed on the very first trial, so P(X = 1) = 1 and the mean is also 1. As p approaches 0, the expected number of trials grows very large and the distribution becomes more spread out. A value of p = 0 is not valid because success would never occur.

Use the geometric distribution when you are interested in how many trials are needed to achieve the first success, rather than how many successes occur in a fixed number of trials (which is the binomial setting). For example, 'How many times must I roll a die before getting a 6?' is a geometric question, while 'How many 6s appear in 10 rolls?' is a binomial question.