# Geometric Probability Calculator

Free geometric probability calculator. Find the probability of first success on trial k, cumulative probabilities, expected value, and variance for the geometric distribution.

## What this calculates

Calculate the probability that the first success occurs on a specific trial. Enter the probability of success per trial and the trial number to get exact and cumulative probabilities.

## Inputs

- **Probability of Success (p)** — min 0.0001, max 1 — The probability of success on each independent trial (between 0 and 1).
- **Trial Number (k)** — min 1, max 1000 — The trial on which you want the first success to occur.

## Outputs

- **P(X = k)** — Probability of first success on exactly trial k.
- **P(X ≤ k)** — Probability of first success on or before trial k.
- **P(X > k)** — Probability that more than k trials are needed.
- **Expected Value E(X)** — Average number of trials until first success (1/p).
- **Variance** — Variance of the geometric distribution ((1-p)/p²).
- **Standard Deviation** — Standard deviation of the distribution.
- **Formula** — formatted as text — The probability formula used.

## Details

The geometric distribution models the number of independent Bernoulli trials needed to get the first success.

**Probability Mass Function:**
P(X = k) = (1 - p)^(k-1) * p

This gives the probability of (k-1) failures followed by one success.

**Cumulative Distribution Function:**
P(X ≤ k) = 1 - (1 - p)^k

This gives the probability that the first success happens on or before trial k.

**Key Properties:**

- **Expected value:** E(X) = 1/p (average trials until first success)
- **Variance:** Var(X) = (1 - p) / p²
- **Memoryless property:** The probability of success on the next trial is always p, regardless of how many failures have occurred

**Examples:**

- Rolling a 6 on a die: p = 1/6, expected 6 rolls on average
- Making a free throw (80% shooter): p = 0.8, expected 1.25 shots
- Finding a defective item (1% defect rate): p = 0.01, expected 100 items

## Frequently Asked Questions

**Q: What is the memoryless property?**

A: The geometric distribution is the only discrete distribution with the memoryless property. This means the probability of waiting k more trials for success is the same regardless of how many trials have already failed. If you have flipped tails 10 times, the probability of heads on the next flip is still p.

**Q: What is the difference between geometric and binomial distributions?**

A: The binomial distribution counts the number of successes in a fixed number of trials. The geometric distribution counts the number of trials until the first success. Binomial has a fixed n; geometric has a random n that depends on when the first success occurs.

**Q: Can p equal 0 or 1?**

A: If p = 1, success is guaranteed on every trial, so P(X = 1) = 1 and the distribution is trivial. If p = 0, success never occurs and the expected value would be infinite. This calculator requires p to be greater than 0 and at most 1.

**Q: Why are there two definitions of the geometric distribution?**

A: Some textbooks define X as the trial on which the first success occurs (X = 1, 2, 3, ...). Others define X as the number of failures before the first success (X = 0, 1, 2, ...). This calculator uses the first definition, where X = k means success on trial k.

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Source: https://vastcalc.com/calculators/statistics/geometric-probability
Category: Statistics
Last updated: 2026-04-08