Geometric Distribution Calculator

Calculate PMF, CDF, mean and variance for the geometric distribution (trials until first success).

About Geometric Distribution Calculator

The geometric distribution models the number of independent Bernoulli trials needed to get the first success. If p is the probability of success on each trial, then X counts which trial yields the first success (X ≥ 1).

Formula

P(X=k) = p × (1−p)^(k−1), for k = 1, 2, 3, ...

p: Success probability per trial
k: Trial number of first success

Worked example

With the default values below, Geometric Distribution Calculator returns P(X = k) of 0.10290000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Success Probability (p)accepts 0–1: 0.3
Trial number of first success (k ≥ 1)minimum 1: 4

Results

P(X = k): 0.10290000
P(X ≤ k): 0.75990000
Mean (1/p): 3.3333
Variance ((1−p)/p²): 7.7778

Step by step

1Values used
2Formula applied
3P(X = k)= 0.10290000
4P(X ≤ k)= 0.75990000
5Mean (1/p)= 3.3333
6Variance ((1−p)/p²)= 7.7778

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is the memoryless property?

The geometric distribution is the only discrete distribution that is memoryless: no matter how many failures have occurred, the probability of success on the next trial remains p.

How does this differ from the binomial?

The binomial counts successes in a fixed number of trials. The geometric counts trials until the first success — the number of trials is the random variable, not the number of successes.