Geometric Distribution Calculator
Evaluate probabilities for the number of failures before the first success in independent Bernoulli trials.
P(X = k): 0.0720
P(X ≤ k): 0.8319
P(X > k): 0.1681
Mean: 2.333
Variance: 7.778
Mode: 0
How to Use This Calculator
- Enter the probability of success in each independent trial.
- Specify the number of failures before the first success.
- Review the probability mass, cumulative probabilities, and moments.
- Use results to describe waiting times or count trials until success scenarios.
Formula
P(X = k) = (1 − p)k · p
P(X ≤ k) = 1 − (1 − p)k + 1
Mean = (1 − p) / p • Variance = (1 − p) / p²
This formulation counts failures before the first success (support k = 0, 1, 2, …). For counting trials until success (support k ≥ 1), shift by one.
Frequently Asked Questions
Is this the same as geometric waiting time?
Yes. This version counts failures before success. Add 1 to obtain the number of trials until success.
When is the geometric model appropriate?
When trials are independent, have identical success probability, and you seek the first success.
How do I model multiple successes?
Use the negative binomial distribution to model waiting time until r successes occur.