Geometric distribution

The geometric distribution models the number of independent trials needed for the first success, with probability (1-p)^(k-1) times p on trial k, and a mean of 1/p trials.

ByHoang TruongUpdated July 11, 2026

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A sales rep closes a cold call with probability p = 0.20. The chance the first sale lands on exactly the third call is P(X=3) = 0.80² × 0.20 = 0.128, or 12.8%: calls one and two fail, then call three succeeds. The mean number of calls needed for a first sale is 1 ÷ 0.20 = 5 calls.

Where it fits

TopicProbability & DistributionsAdvancedSubjectData Analysis & StatisticsAdvanced

The formula

LaTeX

P(X=k) = (1-p)^{k-1}p

Variables

$X$: Number of trials until the first success
$k$: Trial number of interest
$p$: Probability of success on each trial

Gives the probability that the first success occurs exactly on the k-th independent trial, given a constant per-trial success probability p.

LaTeX

E[X] = \frac{1}{p}

Variables

$E [X]$: Mean number of trials until the first success
$p$: Probability of success on each trial

Gives the average number of trials needed to reach the first success.

Check yourself

PracticeCORE

An online ad is clicked by any given visitor with probability p = 0.10, independently across visitors. What is the probability that the first click happens on exactly the fourth visitor?

Select an answer to check your understanding.