What Is Geometric Distribution?
The geometric distribution describes how many independent trials you need until the first success, when each trial has the same success probability p. The probability that the first success is on trial k is (1 − p)^(k − 1) × p.
The calculator returns that probability, the cumulative probability, the expected number of trials 1/p, the expected number of failures before the first success, the median, the standard deviation and the number of trials needed for a 95 % chance. The distribution is memoryless, so past failures do not change the chance on the next trial.
Below you will find the formulas, a variables table, a worked example and answers to common questions such as the two conventions for counting trials versus failures.
How to Use the Geometric Distribution Calculator
- Enter the probability of success on a single trial.
- Enter the trial number k you are interested in.
- Pick whether you want exactly k, up to k, before k, k or later, or more than k.
- Read the expected number of trials and the number needed for a 95% chance, then check the charts.
Geometric Distribution Formula
Probability first success on trial k
Cumulative probability
Mean and variance
Memoryless property
Geometric Distribution Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| p | Success probability per trial | 0 < p ≤ 1 |
| k | Trial number of the first success | 1, 2, 3, … |
| P(X = k) | Probability of first success on trial k | (1 − p)^(k − 1) p |
| E[X] | Expected number of trials | 1 ÷ p |
Geometric Distribution Worked Example
With the example values loaded in the calculator above, the steps are:
- The probability the first success happens on trial 4 is P(X = k) = (1 − p)^(k − 1) × p = 0.8^3 × 0.2 = 0.1024.
- Cumulative: P(X ≤ 4) = 1 − (1 − p)^4 = 1 − 0.4096 = 0.5904.
- Mean = 1 ÷ p = 5; variance = (1 − p) ÷ p² = 20.
Result: P(X = 4) = 0.1024; Expected number of trials 1/p = 5.
How to Read a Geometric Probability
The geometric distribution counts the trials up to and including the first success, when each trial has the same chance p. The mean number of trials is 1 ÷ p and the variance is (1 − p) ÷ p². A success chance of 0.2 means you expect 5 trials on average, but the spread is large.
The distribution is memoryless. Having already failed many times does not make the next trial any more likely to succeed. The calculator shows P(X = k), the chance of success within k trials and the chance of none.
Two versions of the geometric distribution
Textbooks define it in two ways, and mixing them is the most common source of wrong answers. One counts the trial on which the first success happens (k = 1, 2, 3, …), the version used here. The other counts the number of failures before the first success (k = 0, 1, 2, …). They are the same distribution shifted by one.
| Trials until first success | Failures before first success | |
|---|---|---|
| Support | 1, 2, 3, … | 0, 1, 2, … |
| P(X = k) | (1 − p)^(k−1) p | (1 − p)^k p |
| Mean | 1/p | (1 − p)/p |
| Variance | (1 − p)/p² | (1 − p)/p² |
The memoryless property
The geometric distribution has no memory. If you have already failed ten times, the chance of success on the next trial is still p, and the expected additional wait is still 1/p. A run of bad luck does not make success “due”. It is the discrete analogue of the exponential distribution and the only discrete distribution with this property.
Worked example
A salesperson closes 20% of calls. The probability that the first sale occurs on the 4th call is 0.8³ × 0.2 = 0.1024. The expected number of calls until the first sale is 1 ÷ 0.2 = 5, with a standard deviation of about 4.5. To be 95% sure of at least one sale you need 14 calls, because 1 − 0.8¹⁴ ≈ 0.956.
Conditions
- Trials are independent.
- Each trial has only two outcomes, success or failure.
- The probability p is constant from trial to trial.
- You stop at the first success.
Where Geometric Distribution Is Used
First success
Find the chance the first success comes on trial k.
Within k trials
Find the chance of at least one success in k trials.
Expected trials
Read the expected number of trials and failures.
Planning attempts
Work out how many tries give a chosen chance of success.
Common Geometric Distribution Mistakes to Avoid
- Check whether your course counts the trial of the first success or the failures before it. The calculator shows both.
- Trials must be independent with the same p.
- The distribution is right-skewed, so the mean can be well above the median.
- For "at least one success in k trials" use 1 − (1 − p)ᵏ.
Geometric Distribution Terms and Definitions
- Memoryless
- Past failures do not change the chance of future success.
- Geometric series
- The sum that gives the cumulative formula 1 − (1 − p)ᵏ.
Geometric Distribution FAQs
What is the geometric distribution?
The distribution of the number of independent trials needed to get the first success when each trial succeeds with the same probability p.
What is the mean of a geometric distribution?
1/p for the trials version. If p = 0.2, you expect 5 trials.
What is the difference between geometric and binomial?
Binomial counts successes in a fixed number of trials. Geometric counts trials until the first success.
What does memoryless mean?
Having already failed several times does not change the probability of success on the next trial.
How do I find the probability that it takes more than k trials?
P(X > k) = (1 − p)^k, the chance that the first k trials all fail.
How many trials do I need for 95% confidence?
The smallest n with 1 − (1 − p)ⁿ ≥ 0.95, shown in the results.