Geometric Distribution Calculator
Use our geometric distribution calculator to compute geometric distribution probability quickly. Includes the geometric distribution definition, formula, properties (including the memoryless property), mean and variance formulas, expected value and variance, and guidance on when to use geometric distribution in stats.
What is Geometric Distribution?
The geometric distribution is a discrete probability distribution that models the number of trials needed to get the first success in repeated independent Bernoulli trials (like repeated coin flips).
If you’re asking what is geometric distribution, it’s used when every trial has the same probability of success p, trials are independent, and you’re counting how many trials happen until the first success occurs.
Geometric distribution shows up in many courses (including geometric distribution AP stats). You might also see unrelated searches like geometric distribution AP human geography or geometric distribution pattern, but in statistics the term specifically refers to this probability model.
Geometric Distribution Formula
There are two common conventions: counting the trial number of the first success (k = 1, 2, 3, …) or counting failures before the first success (k = 0, 1, 2, …). This calculator uses the “trial number of first success” form by default.
Valid for k = 1, 2, 3, … and 0 < p ≤ 1.
Geometric distribution mean formula (trial-count version).
Geometric distribution expected value and variance formulas.
Probability the first success happens on the 3rd trial.
Mean is 1/p and variance is (1-p)/p^2.
How to Use the Geometric Distribution Calculator
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Enter the probability of success p (between 0 and 1).
- 2
Enter k, the trial number you want (first success on trial k).
- 3
The calculator computes geometric distribution probability using P(X = k) = (1 - p)^(k - 1) · p.
- 4
Optionally, view the geometric distribution expected value and variance: E[X] = 1/p and Var(X) = (1 - p)/p^2.
Frequently Asked Questions
A discrete distribution modeling the number of trials until the first success in repeated independent trials with constant success probability p.
Discrete.
Use it when trials are independent, each trial has the same success probability p, and you’re counting trials until the first success.
Pick p and k, then compute P(X=k) = (1-p)^(k-1)·p (trial-count convention).
Key properties include being discrete, having mean 1/p, variance (1-p)/p^2, and the memoryless property.
Yes. Memoryless means P(X > m + n | X > m) = P(X > n). Past failures don’t change the future distribution when p stays constant.
E[X] = 1/p (for the trial-count version).
E[X] = 1/p and Var(X) = (1-p)/p^2 (trial-count version).
They’re different. Binomial models the number of successes in a fixed number of trials. Geometric models the number of trials until the first success.
It focuses on the geometric PMF, mean/variance, interpreting k as trials until first success, and the memoryless property.
That’s about geometric sequences, not geometric distribution. A geometric sequence uses multiplication by a common ratio (which can be a fraction), so it can look like division when the ratio is less than 1.