Geometric Distribution Calculator
How many attempts until the first success?
Your numbers
Your results
P(first success on trial k)
P(first success on trial k)
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P(success within k trials)
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Expected number of trials
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Geometric Distribution Calculator by TheCalcTools
How the Geometric Distribution Calculator works
Enter the success probability for a single trial and the trial number k.
The first success on trial k requires k − 1 failures then a success.
The expected number of trials is the reciprocal of the probability.
The formula
P(X = k) = (1 − p)ᵏ⁻¹ pP(X ≤ k) = 1 − (1 − p)ᵏ
Worked example
Using the default values shown in the calculator:
| Success probability per trial (%) | 20 |
|---|---|
| Trial number (k) | 3 |
- P(first success on trial 3) = 0.8² × 0.2 = 0.1280.
- P(success within 3 trials) = 1 − 0.8³ = 0.4880.
- Expected trials = 1 ÷ 0.2 = 5.
Result:
| P(first success on trial k) | 0.1280 |
|---|---|
| P(success within k trials) | 0.4880 |
| Expected number of trials | 5.00 |
Frequently asked questions
What is the memoryless property?
The chance of success on the next trial does not depend on how many failures came before.
How can I find the chance of at least one success?
Use the ‘within k trials’ row: one minus the chance that all k trials fail.
Sources and further reading
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This calculator gives estimates for information only. Check important results independently before you rely on them. See our disclaimer.