Quick answer
In a geometric distribution P(X = k) = (1 − p)^(k − 1) p and P(X ≤ k) = 1 − (1 − p)^k. With p = 0.2, the first success on trial 3 has probability 0.128, P(X ≤ 3) = 0.488 and the mean number of trials is 5.
Enter the success probability and the trial number. The calculator returns P(X = k), P(X ≤ k) and the mean.
Related tools: Probability Calculator: Probability of A and B, and A or B, for independent or mutually exclusive events. Binomial Distribution Calculator: Binomial probabilities for n trials. Sampling Distribution of the Mean Calculator: Probability for a sample mean using the central limit theorem.
How to use the Geometric Probability Calculator
- Enter success probability (between 0 and 1).
- Enter trial number (a whole number, 1 or more).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Success probability | A number | > 0, ≤ 1 |
| Trial number | A whole number | ≥ 1 |
Formula and method
Formula
P(X = k) = (1 − p)^(k − 1) p; P(X ≤ k) = 1 − (1 − p)^k
IEEE-754 double precision; shown to at most 6 decimal places.
Geometric distribution on k = 1, 2, 3, …
Worked examples
This example is verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Success probability = 0.2; Trial number = 3.
| Result | Value |
|---|---|
| P(X = k) | 0.128 |
| P(X ≤ k) | 0.488 |
| Mean trials | 5 |
The first line is the headline result (p(X = k)). Below it the geometric Probability Calculator lists p(X ≤ k) and mean trials, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Worked example
Two failures then a success: 0.8 × 0.8 × 0.2 = 0.128. The chance of a success within 3 trials is 1 − 0.8³ = 0.488. The expected wait is 1 ÷ p = 5 trials.
Memoryless
The distribution is memoryless: past failures don't change the chance of success on the next trial. It models waiting for the first six on a die (p = 1/6, mean 6) or the first defect. p must satisfy 0 < p ≤ 1.
How the result changes with success probability
Holding every other input at the first worked example, this table changes only success probability. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Success probability | P(X = k) | P(X ≤ k) | Mean trials |
|---|---|---|---|
| 0.1 | 0.081 | 0.271 | 10 |
| 0.2 | 0.128 | 0.488 | 5 |
| 0.4 | 0.144 | 0.784 | 2.5 |
| 0.8 | 0.032 | 0.992 | 1.25 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Success probability: entering "0" shows "Success probability must be greater than 0." beside that field and no result.
Choosing between the Geometric Probability Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Probability Calculator](/statistics/probability-calculator/) | p, p and events are | P(A and B) |
| [Binomial Distribution Calculator](/statistics/binomial-distribution/) | trials, success probability and successes | P(X = 5) |
| [Sampling Distribution of the Mean Calculator](/statistics/normal-probability-sampling-distributions/) | population mean, population SD and sample size | P(x̄ < value) |
| [IQ Percentile Calculator](/statistics/iq-percentile/) | iQ score, mean and standard deviation | Percentile |
| [Empirical Rule Calculator](/statistics/empirical-rule/) | mean and standard deviation | ≈68% (μ ± σ) |
Background: probability and distributions
A probability model only applies when its conditions hold. The binomial model needs a fixed number of independent trials with two outcomes and a constant success probability; its mass function gives the chance of exactly k successes. The normal density is f(x) = 1 ÷ (σ√(2π)) × e^(−(x − μ)² ÷ (2σ²)).
A cumulative probability adds up everything at or below a value, a density or mass function gives the value at a point. For a standard normal variable, z = −1.96 leaves 0.025 in the lower tail, which is why ±1.96 bounds a 95% interval.
Assumptions and limitations
- 0 < p ≤ 1
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for success probability to confirm this.
Frequently asked questions
What is the mean of a geometric distribution?
The mean number of trials until the first success is 1 ÷ p; with p = 0.2 that is 5.
What is the variance?
The variance is (1 − p) ÷ p²; with p = 0.2 that is 0.8 ÷ 0.04 = 20.
How does it differ from binomial?
Geometric counts trials until the first success.
What does the Geometric Probability Calculator calculate?
Probability the first success occurs on trial k. It returns P(X = k), P(X ≤ k) and Mean trials.
What formula does the Geometric Probability Calculator use?
P(X = k) = (1 − p)^(k − 1) p; P(X ≤ k) = 1 − (1 − p)^k. Geometric distribution on k = 1, 2, 3, …
What do I need to enter in the geometric Probability Calculator?
You enter Success probability and Trial number.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. 0 < p ≤ 1.
What is an example of a geometric Probability result?
With success probability 0.2 and trial number 3, the p(X = k) is 0.128.
Related calculators
- Probability Calculator — Probability of A and B, and A or B, for independent or mutually exclusive events.
- Binomial Distribution Calculator — Binomial probabilities for n trials.
- Sampling Distribution of the Mean Calculator — Probability for a sample mean using the central limit theorem.
- IQ Percentile Calculator — Percentile of an IQ score on a normal scale.
- Empirical Rule Calculator — 68–95–99.7 intervals for a normal distribution.
- Poisson Distribution Calculator — Poisson probabilities for a given mean rate.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods (itl.nist.gov/div898/handbook)
- Standard textbook definitions (e.g. Moore, McCabe & Craig, Introduction to the Practice of Statistics)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.