Geometric Probability Calculator
Find the probability that the first success occurs on exactly the nth trial, given a constant success probability per trial. Use it for modeling first-time events like the first defective item on a production line or the first sale made by a salesperson.
About this calculator
The geometric distribution models the number of independent Bernoulli trials needed to achieve the first success, where each trial has a constant probability of success p. The probability mass function is P(X = n) = (1 − p)^(n−1) × p, where n is the trial number on which the first success occurs, p is the probability of success on each trial, and (1 − p) is the probability of failure. The factor (1 − p)^(n−1) accounts for the fact that the first n−1 trials must all be failures before the success on trial n. The distribution is memoryless, meaning the probability of success on the next trial does not depend on how many failures have already occurred. The mean number of trials until the first success is 1/p, and the distribution is always right-skewed, with the highest probability at n = 1.
How to use
A free-throw shooter makes 70% of attempts, so p = 0.70. What is the probability the first successful shot occurs on exactly the 3rd attempt (n = 3)? Apply the formula: P(X = 3) = (1 − 0.70)^(3−1) × 0.70 = (0.30)² × 0.70 = 0.09 × 0.70 = 0.063. There is a 6.3% chance that the athlete misses the first two shots and makes exactly the third. The highest probability remains at n = 1: P(X = 1) = (0.30)⁰ × 0.70 = 0.70.
Frequently asked questions
What does the memoryless property of the geometric distribution mean in practice?
The memoryless property means that past failures carry no information about future outcomes. If you have already failed 5 times, the probability of succeeding on the very next attempt is still exactly p — the same as it was on attempt number one. This holds because each trial is independent. In practice, this means you should not expect that a success is 'due' after a long streak of failures. The property is unique to the geometric (discrete) and exponential (continuous) distributions and is why they model processes where no wear, learning, or accumulation occurs between trials.
How is the geometric distribution different from the negative binomial distribution?
The geometric distribution is a special case of the negative binomial distribution. The geometric distribution asks: how many trials until the first success? The negative binomial distribution generalizes this to: how many trials until the rth success? When r = 1, the negative binomial reduces exactly to the geometric distribution. Use the geometric distribution when you only care about the very first success, and use the negative binomial when you need the rth success for any r greater than one.
When is the geometric probability distribution appropriate for modeling real-world events?
The geometric distribution is appropriate when you are counting independent, identical trials until the first success, with a constant probability of success on each attempt. Real-world examples include the number of customer calls needed to make the first sale, the number of parts inspected before finding the first defect, or the number of coin flips until the first heads. It is not appropriate when the probability of success changes over time (e.g., a learning effect), when trials are not independent, or when you are counting total successes rather than the position of the first success.