Geometric Distribution Calculator
Probability that the first success lands on trial k.
Enter your values
ⓘ Live results
Results
- P(first success on trial k)
- 10.5469%
- P(success within k trials)
- 68.3594%
- P(more than k trials needed)
- 31.6406%
- Expected number of trials
- 4
- Standard deviation
- 3.464
Quick answer
For independent trials with success probability p, the chance the first success is on trial k is (1 − p)^(k−1) × p, and the expected wait is 1 ÷ p trials.
P(X = k) = (1 − p)^(k−1) · p
How to use the Geometric Distribution Calculator
- 1Enter the chance of success on a single trial as a decimal.
- 2Enter the trial number you are asking about.
- 3Compare the exact probability with the cumulative chance of succeeding by then.
Worked example
Using the values the calculator loads with:
Inputs
- Probability of success per trial0.25
- Trial number k4
Results
- P(first success on trial k)10.5469%
- P(success within k trials)68.3594%
- P(more than k trials needed)31.6406%
- Expected number of trials4
- Standard deviation3.464
Frequently asked questions
What is the difference from the binomial distribution?
Binomial fixes the number of trials and counts successes. Geometric fixes one success and counts the trials it takes.
Why is the mean 1 ÷ p?
If something works one time in four, you expect to wait four attempts — the intuition matches the algebra exactly.
About this calculator
- Built and reviewed by James Whitfield · How we build and test calculators
- Report an error
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.