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🎯 Geometric Distribution Calculator

Calculate the probability that the first success occurs on trial k. Computes P(X=k), cumulative P(X≤k), mean, and variance for a geometric distribution.

Between 0 and 1

Must be ≥ 1

Geometric Distribution Formulas

PMF (first success on trial k)
P(X = k) = (1 − p)^(k−1) × p
CDF
P(X ≤ k) = 1 − (1 − p)^k
Mean and Variance
E[X] = 1/p
Var(X) = (1−p) / p²

Real-World Examples

  1. 1
    Free Throw Shooting
    A basketball player with 70% free throw accuracy: what is the probability their first miss happens on the 4th attempt?
  2. 2
    Sales Calls
    A salesperson closes a deal with probability 0.20 per call. What is the expected number of calls until the first sale?
  3. 3
    Defective Items
    An assembly line produces 1% defective items. What is P(first defect found on the 5th inspection)?
  4. 4
    Clinical Trials
    A drug has 40% response rate. What is P(first responder is the 3rd patient)?

Frequently Asked Questions

The geometric distribution models the number of independent Bernoulli trials needed to get the first success, where each trial has probability p of success. It is memoryless — past failures do not affect future probabilities.

The memoryless property means P(X > m + n | X > m) = P(X > n). In plain terms, if you have already had k failures, the distribution of remaining trials until success is the same as at the start. This is unique to the geometric (and exponential) distributions.

The geometric distribution is a special case of the negative binomial distribution with r = 1. The negative binomial models the number of trials until r successes, while geometric models trials until 1 success.

Use it when you are counting the number of trials until the first success, each trial is independent, and the success probability p is constant across trials. Examples include coin flips until heads, or inspections until finding a defect.

P(X = k) = (1−p)^(k−1) × p means there were exactly k−1 failures (each with probability 1−p) before the first success (probability p) on trial k.

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