Geometric Probability Calculator
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.
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