Geometric Distribution Calculator

For repeating an attempt until the first success, with the same chance each time.

First success on this trial

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Share sends a link that reopens these numbers. Copy details pastes the full breakdown as a list.

First success by this trial —

No success yet after this trial —

Mean number of trials —

Standard deviation —

Reading this —

The formula

P(X = k) = (1 − p)^(k − 1) × p P(X ≤ k) = 1 − (1 − p)^k mean = 1 ÷ p

k counts trials up to and including the first success.

The most likely single outcome is always success on the first try, even when the mean is much later.

What is a worked example?

With a 0.2 chance each time, first success on trial 3 has probability 0.128, and by trial 3 it is 0.488.

How many trials on average?

1 ÷ p, which is 5 trials at 0.2. The discrete version of the “no memory” property applies: past failures do not change the next chance.