Geometric Distribution Calculator
Probability that the first success lands on trial k.
Enter your values
Enter the probability of success per trial you are working with.
Enter the trial number k you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- 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
At a glance
| What it does | Probability that the first success lands on trial k. |
|---|---|
| Category | Statistics |
| Inputs needed | Probability of success per trial, Trial number k |
| Main output | P(first success on trial k) |
| Formula | P(X = k) = (1 − p)^(k−1) · p |
| Cost | Free — no sign-up, no download |
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.
Inputs explained
- Probability of success per trial
- Enter the probability of success per trial you are working with.
- Trial number k
- Enter the trial number k you are working with.
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.
What do I need to enter into the Geometric Distribution Calculator?
Just 2 values: probability of success per trial, trial number k. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Geometric Distribution Calculator work out the 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. It applies the formula P(X = k) = (1 − p)^(k−1) · p and shows the working so you can check each step by hand.
Is the Geometric Distribution Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.