What is the geometric distribution?

The geometric distribution models the number of independent Bernoulli trials needed to get the first success, where each trial succeeds with probability p. PMF: P(X = k) = (1 − p)^(k−1) × p, for k = 1, 2, 3, …

What is the difference between the two conventions?

Convention 1 (used here by default) counts the total number of trials until the first success, so k ≥ 1. Convention 2 counts only the failures before the first success, so k ≥ 0. The formulas differ by a shift of 1.

What are the mean and variance of the geometric distribution?

For the "number of trials" convention: mean = 1/p, variance = (1 − p) / p². For example, if p = 0.25 the mean is 4 trials and the variance is 12.

What does the CDF tell me?

The CDF P(X ≤ k) gives the probability that the first success occurs on or before trial k. It equals 1 − (1 − p)^k for the number-of-trials convention.

No. p = 0 means success never occurs (undefined distribution) and p = 1 means the first trial always succeeds. The calculator enforces 0 < p < 1.

Is the geometric distribution memoryless?

Yes. Given that no success has occurred in the first n trials, the distribution of additional trials required is the same as the original. This is the only discrete distribution with this property.

Geometric Distribution Calculator

Name: Geometric Distribution Calculator
Availability: InStock
Author: Nham Vu

Enter a success probability and trial number to compute PMF, CDF, mean, variance, and standard deviation for the geometric distribution.

Parameters

Probability of success p

Must be between 0 and 1 (exclusive).

Trial number k

Trial on which first success may occur (k ≥ 1).

Distribution Statistics

Mean (Expected value): —
Variance: —
Standard Deviation: —

Results for k = 3

PMF — P(X = k)

—

Probability of first success on trial k

CDF — P(X ≤ k)

—

Probability of first success by trial k

P(X > k)

—

Probability first success after trial k

Formula

—

P(X = k) = (1 − p)^(k−1) × p

Probability Table

First 15 trials

k	P(X = k)	P(X ≤ k)

Summary

Enter a success probability and trial number to compute PMF, CDF, mean, variance, and standard deviation for the geometric distribution.

How it works

Enter the probability of success p (a value between 0 and 1).
Enter the trial number k (the trial on which the first success occurs).
The calculator instantly shows P(X = k) (PMF) and P(X ≤ k) (CDF).
Distribution stats — mean, variance, and standard deviation — update automatically.
Use the table to explore probabilities across multiple values of k at once.

Use cases

Finding the probability that the first success happens on exactly the k-th trial.
Quality control: modeling the number of inspections until the first defect is found.
Reliability engineering: estimating the number of attempts before the first failure.
Game design: calculating the chance a player first succeeds on a given turn.
Sales modeling: probability a cold call converts on the n-th attempt.
Network transmission: probability of the first successful packet delivery.

Frequently Asked Questions

Related tools

Complement Probability Calculator Probability Intersection Calculator Odds To Probability Converter

Last updated: 2026-05-23 · Reviewed by Nham Vu