Geometric distribution calculator
Answers questions like: what is the chance my first six comes on the third roll, and how many rolls does it take on average?
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How to use it
- Enter success probability p and trial number k.
- The result appears straight away and updates as you edit.
- Copy the result, or copy a link that keeps the same values to share it.
How it is calculated
P(X = k) = (1 - p)^(k - 1) × p. P(X ≤ k) = 1 - (1 - p)^k. P(X > k) = (1 - p)^k. Mean = 1 ÷ p.
Example
Inputs: Success probability p = 0.2, Trial number k = 3. Result: P(first success on trial k): 0.128; P(success within k trials): 0.488; P(no success in the first k trials): 0.512.
Limits and caveats
- Counts trials up to and including the first success.
- Assumes independent trials with a fixed chance.
Questions about this tool
What is the chance the first six comes on roll 3?
(5/6)^2 × (1/6) ≈ 0.1157, about 11.6%.
Why is the mean 1 ÷ p?
An event with a 20% chance needs 5 tries on average, and 1 ÷ 0.2 = 5.
Does the geometric distribution have memory?
No. After 10 failures the chance on the next try is still p.