Geometric Distribution Calculator
Compute geometric distribution probabilities for the trial on which the first success occurs, along with cumulative probability, mean, and variance.
Probability type
Geometric distribution result
- 1
Failure probability (q)
1 − 0.3 = 0.7 - 2
Failures before trial k
0.7^(3 − 1) = 0.49 - 3
P(X = k)
0.49 × 0.3 = 0.1470Fail the first k−1 trials, then succeed on trial k.
How does this calculator work?
For independent trials each succeeding with probability p, the geometric distribution gives P(X = k) = (1 − p)^(k−1)·p that the first success lands on trial k. Cumulative probability is 1 − (1 − p)^k, the mean wait is 1/p trials, and the variance is (1 − p)/p². Enter p and k to compute each.
Formula
How this is calculated
The geometric distribution models the number of independent Bernoulli trials needed to get the first success, where each trial succeeds with probability p. Enter p as a value between 0 and 1 and k as a positive integer (1, 2, 3, …) indicating the trial of interest. The probability mass function P(X = k) = (1 − p)^(k−1) · p says you must fail the first k − 1 trials (each with probability 1 − p) and then succeed on trial k.
The mode dropdown selects which quantity to report. "Exactly on trial k" returns the PMF. "By trial k" returns the cumulative distribution P(X ≤ k) = 1 − (1 − p)^k, the chance the first success has occurred by trial k. "On or after trial k" returns the survival probability P(X ≥ k) = (1 − p)^(k−1), the chance you are still waiting at the start of trial k. The summary also shows the mean number of trials, 1/p, and the variance, (1 − p)/p².
Assumptions: trials are independent and identically distributed with a constant success probability, and X counts trials starting from 1 (the "shifted" convention). p must satisfy 0 < p ≤ 1; k is rounded to the nearest integer and must be at least 1. As p approaches 0 the expected wait 1/p grows without bound. The bar chart shows the PMF for the first several trials so you can see the geometric decay, with the selected trial highlighted.
Frequently asked questions
This calculator uses the trials-until-first-success convention where k starts at 1, so P(X = 1) = p. If you need the number of failures before the first success (starting at 0), shift k by one.
On average you need 1/p trials to get one success. For example, with p = 0.2 you expect 5 trials. The variance (1 − p)/p² grows quickly as p gets small because rare successes mean long, highly variable waits.
It is the probability that the first success has not happened in the first k − 1 trials, i.e. you are still waiting at trial k. It equals (1 − p)^(k−1), the memoryless survival probability of the geometric distribution.
Also known as
TG we-Calculate Editorial Team. (2026). Geometric Distribution Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/geometric-distribution-calculator
TG we-Calculate Editorial Team. "Geometric Distribution Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/geometric-distribution-calculator.
TG we-Calculate Editorial Team, "Geometric Distribution Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/geometric-distribution-calculator
@misc{wecalculate_geometric_distribution_calculator, title = {Geometric Distribution Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/geometric-distribution-calculator}}, year = {2026}, note = {TG we-Calculate} }
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