How to use it
- Enter the probability of success p on each trial.
- Enter k and choose whether X counts trials (first success on trial k) or failures before the first success.
- Read P(X = k) and the cumulative probabilities. The chart shows the distribution with k highlighted.
Formulas
With q = 1 − p and j = the number of failures before the success (j = k − 1 counting trials, j = k counting failures):
- P(X = k) = qʲ · p
- P(X > k) = q^(j + 1), P(X ≥ k) = qʲ, P(X ≤ k) = 1 − q^(j + 1)
- Mean 1/p (trials) or q/p (failures); variance q/p²
Worked example
A salesperson closes 20 % of calls (p = 0.2). The chance that the first sale is on call 3 is 0.8² · 0.2 = 0.128. The chance of a sale within the first 3 calls is 1 − 0.8³ = 0.488, and the chance of three calls without a sale is 0.512. On average the first sale takes 5 calls, with variance 0.8/0.04 = 20 and standard deviation 4.47214.
Counting failures instead, P(X = 3) is the chance of exactly 3 failed calls before the first sale: 0.8³ · 0.2 = 0.1024.
Tips and common mistakes
- The trials must be independent with the same p; drawing cards without replacement is not geometric.
- “At least one success in n trials” is 1 − (1 − p)ⁿ, the same as P(X ≤ n) when counting trials.
- Check which version your textbook uses before comparing answers; they differ by one in k and in the mean.