What is the geometric distribution?

The geometric distribution models the number of trials needed to get the first success in a sequence of independent Bernoulli trials, each with the same probability of success p. The probability that the first success occurs on the k-th trial is P(X = k) = (1 minus p) to the power (k minus 1) times p, for k = 1, 2, 3 and so on.

What is the mean of the geometric distribution?

The mean (expected value) of the geometric distribution is 1 divided by p. So if each trial has a 30 percent chance of success, you expect on average 1 divided by 0.3 = 3.33 trials until the first success. The variance is (1 minus p) divided by p squared.

Geometric Distribution Calculator

The geometric distribution answers a simple but important question: if you repeat an experiment with a fixed probability of success on each attempt, what is the chance that your first success comes on exactly the k-th try? Each attempt is independent and has the same probability p of succeeding. A free-throw shooter with a 30 percent success rate, a quality control inspector checking items until finding the first defect, or a student re-sitting a test until they pass: all of these follow the geometric distribution. The probability that the first success occurs on trial k is P(X = k) = (1 minus p) to the power (k minus 1) times p. The cumulative probability P(X less than or equal to k) = 1 minus (1 minus p) to the power k gives the chance of succeeding by trial k. The expected number of trials to first success is 1 divided by p. This calculator takes the success probability p (between 0 and 1) and the trial number k (a positive integer), and returns the exact probability P(X = k), the cumulative probability P(X less than or equal to k), the probability of still waiting P(X greater than k), and the distribution mean, variance and standard deviation. It is useful for students studying probability theory, quality engineers, sports analysts and anyone dealing with repeated Bernoulli trials.

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Success probability (p)

Trial number (k)

0.1029

P(X = 4): first success on trial 4

P(X ≤ 4)0.7599

P(X > 4)0.2401

Mean (1/p)3.33

Variance7.78

Std Dev2.79

P(X = k) = (1-p)^(k-1) times p. P(X ≤ k) = 1 - (1-p)^k. p must be between 0 and 1 exclusive. k must be a positive integer.

How it works

For a geometric distribution with success probability p and trial number k:
P(X = k) = (1-p)^k-1 × p
P(X ≤ k) = 1 - (1-p)^k
P(X > k) = (1-p)^k
Mean = 1/p
Variance = (1-p) / p²
Standard deviation = √(variance)
The distribution has the memoryless property: the number of additional trials needed is independent of how many trials have already failed.

Worked example

With p = 0.3 and k = 4: P(X = 4) = (0.7)³ × 0.3 = 0.343 × 0.3 = 0.1029. P(X ≤ 4) = 1 - (0.7)⁴ = 1 - 0.2401 = 0.7599. P(X > 4) = (0.7)⁴ = 0.2401. Mean = 1/0.3 = 3.33. Variance = 0.7/0.09 = 7.78. Std dev = √7.78 = 2.79.

Related calculators

Exponential Distribution Calculator: the continuous analogue of the geometric distribution.
Effect Size Calculator: measure the practical significance of a statistical result.
Percent Error Calculator: compare a computed probability to a reference value.
Measurement Uncertainty Calculator: quantify the range of possible values around repeated measurements.

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