The geometric distribution
The geometric distribution describes how long you wait for the first success in a series of independent trials, each with the same probability of success p. How many sales calls until the first sale? How many rolls until a six? How many attempts until a component fails?
There are two common conventions, and the calculator handles both:
- number of trials until the first success, X = 1, 2, 3, … (the default);
- number of failures before the first success, Y = 0, 1, 2, …, so Y = X − 1.
The formulas (trials convention)
- P(X = k) = (1 − p)^(k − 1) × p;
- P(X ≤ k) = 1 − (1 − p)^k;
- P(X > k) = (1 − p)^k;
- mean 1 ÷ p, variance (1 − p) ÷ p².
Example
Each trial succeeds with probability p = 0.2:
- first success exactly on trial 3: 0.8² × 0.2 = 0.128;
- first success by trial 3: 1 − 0.8³ = 0.488;
- no success in the first 3 trials: 0.512;
- mean of 5 trials, variance 20, median 4;
- 11 trials are needed for a 90% chance of at least one success.
The memoryless property
The geometric distribution is memoryless: if no success has happened after m trials, the number of extra trials needed follows the same distribution as at the start. Ten failed rolls do not make a six more likely on the next roll. Believing otherwise is the gambler’s fallacy. The exponential distribution is its continuous counterpart, with the same property.
Common uses
- Reliability: the number of cycles before a first failure, when each cycle carries the same small failure risk.
- Games and lotteries: expected number of tickets or rolls before a win.
- Marketing: calls or emails before a first conversion.
- Quality control: items inspected before the first defect.
“At least once” probability
The chance of at least one success in n trials is 1 − (1 − p)ⁿ. With p = 1% per trial, 69 trials give a 50% chance and 230 give 90%. That is often far more than intuition suggests.