What Is Geometric Distribution?
The geometric distribution models the number of trials needed to obtain the first success in a sequence of independent trials, each with the same success probability p. A common version counts the number of failures k that occur before that first success.
Because the trials are independent, the probability of seeing exactly k failures and then a success on the next trial is (1 − p) raised to the power k, times p. This distribution is the discrete waiting-time partner to the binomial distribution, which instead counts successes in a fixed number of trials.
Formula
P(X = k failures before first success) = (1 − p)ᵏ × p
where:
k = number of failures before the first success (k ≥ 0)
p = probability of success on each trial
Applications
- Modeling how many calls a salesperson makes before the first sale
- Estimating the number of defective checks before a machine passes
- Counting free throws missed before the first successful basket
- Describing the waiting time until a rare event first occurs in a process
Sources
- Moore, D. S., McCabe, G. P., & Craig, B. A. (2021). Introduction to the Practice of Statistics (10th ed.). Macmillan.
- Montgomery, D. C., & Runger, G. C. (2018). Applied Statistics and Probability for Engineers (7th ed.). Wiley.