Geometric Distribution Calculator (Trials to First Success)
By Santaji GadepatilinUpdated November 4, 2026
The geometric distribution describes how many independent tries it takes to get the first success when each try has the same probability p. This geometric distribution calculator gives the probability that the first success comes exactly on trial k, by trial k or only after k, along with the expected number of trials.
How to use the geometric distribution calculator
Enter the probability of success on each trial and the trial number k. The calculator returns the probability that the first success happens exactly on trial k, by trial k at the latest, and not within k trials, plus the mean and standard deviation of the number of trials. Clear p and enter a target P(X ≤ k) to find the success rate needed.
Geometric distribution formulas
P(X = k) = (1 − p)k−1 p · P(X ≤ k) = 1 − (1 − p)k
μ = 1 ÷ p · σ = √(1 − p) ÷ p
Example
A sales call succeeds 20% of the time. The chance that the first sale comes on the 3rd call:
- P(X = 3) = 0.8² × 0.2 = 12.8%
- P(X ≤ 3) = 1 − 0.8³ = 48.8%, P(X > 3) = 0.8³ = 51.2%
- Expected calls = 1 ÷ 0.2 = 5, standard deviation = √0.8 ÷ 0.2 = 4.4721
Geometric probabilities for p = 20%
| k | P(X = k) | P(X ≤ k) | P(X > k) |
|---|---|---|---|
| 1 | 20.00% | 20.00% | 80.00% |
| 2 | 16.00% | 36.00% | 64.00% |
| 3 | 12.80% | 48.80% | 51.20% |
| 4 | 10.24% | 59.04% | 40.96% |
| 5 | 8.19% | 67.23% | 32.77% |
| 6 | 6.55% | 73.79% | 26.21% |
| 7 | 5.24% | 79.03% | 20.97% |
| 8 | 4.19% | 83.22% | 16.78% |
| 9 | 3.36% | 86.58% | 13.42% |
| 10 | 2.68% | 89.26% | 10.74% |
| 11 | 2.15% | 91.41% | 8.59% |
| 12 | 1.72% | 93.13% | 6.87% |
Notes
- The trials must be independent with a constant success probability.
- To count successes in a fixed number of trials instead, use the binomial probability calculator.
Frequently asked questions
What is the geometric distribution formula?
P(X = k) = (1 − p)^(k − 1) × p — k − 1 failures followed by one success. The probability of success within k trials is P(X ≤ k) = 1 − (1 − p)^k.
What is the mean of a geometric distribution?
The expected number of trials up to and including the first success is 1 ÷ p, and the variance is (1 − p) ÷ p². With p = 20% you expect 5 trials on average.
What is the difference between the two definitions of the geometric distribution?
This calculator counts trials up to and including the first success (k = 1, 2, 3 …). Some textbooks count only the failures before it (k = 0, 1, 2 …); then P = (1 − p)^k × p and the mean is (1 − p) ÷ p.
Is the geometric distribution memoryless?
Yes — having already failed several times does not change the chance of success on the next trial. It is the discrete counterpart of the exponential distribution.