Geometric Distribution Calculator

The geometric distribution models how many independent trials you need before getting the first success. Each trial has the same probability p of success.

If X is the number of trials until (and including) the first success:

P(X = k) = (1-p)^(k-1) × p

This reads: k-1 failures each with probability (1-p), then one success with probability p.

Cumulative: P(X ≤ k) = 1 - (1-p)^k Survival: P(X > k) = (1-p)^k

Mean: E[X] = 1/p

Variance: Var(X) = (1-p)/p²

Standard deviation: σ = √((1-p))/p

Example uses

How many coin flips until heads? p = 0.5, so on average 2 flips. P(X=1) = 0.5, P(X=2) = 0.25, P(X=3) = 0.125.

Manufacturing defects: a process produces 2% defective items. How many items inspected until finding one defect? Mean = 1/0.02 = 50 items.

The memoryless property

The geometric distribution is memoryless: P(X > n+m | X > n) = P(X > m). Past failures provide no information about future outcomes. If you have already flipped 10 tails, the probability of needing 5 more flips is exactly the same as if you had just started. This is the discrete analog of the exponential distribution’s memoryless property.

Two conventions

Some textbooks define X as the number of failures before the first success (so X can be 0). This gives P(X=k) = (1-p)^k × p. The calculator uses the more common convention where X = total trials (minimum 1).

To switch: if you want the number of failures, subtract 1 from any result. Mean failures = (1-p)/p, mean trials = 1/p.