Binomial Probability Calculator

Binomial probability formula

P(X = k) = C(n, k) × p^k × (1 − p)^(n − k)

Use it when there is a fixed number of independent trials, each with only two outcomes and the same probability of success.

Worked example

Flipping a fair coin 10 times, the chance of exactly 3 heads is C(10, 3) × 0.5¹⁰ = 120 ÷ 1,024 = 0.1172. The chance of at most 3 heads is 176 ÷ 1,024 = 0.1719.

Mean and spread

Mean = n × p Variance = n × p × (1 − p)

Frequently asked questions

When should I use a binomial distribution?

When you count successes in a fixed number of independent yes/no trials with the same success probability.

What is the difference between P(X ≤ k) and P(X < k)?

P(X ≤ k) includes exactly k successes; P(X < k) does not.

How do I find the probability of at least one success?

Use 1 − P(X = 0) = 1 − (1 − p)^n.

Can I use the normal approximation?

For large n where np and n(1 − p) are both at least about 10, the normal distribution gives a good approximation.