Binomial Distribution Calculator
Find binomial probabilities: the chance of exactly, at most or at least k successes in n independent trials.
About the Binomial Distribution Calculator
The binomial distribution counts successes in a fixed number of independent yes/no trials with the same success probability: heads in coin flips, defective items in a batch, or people who click an ad.
“Fit to observed data” is the engineering-mathematics exercise: from a frequency table of 0, 1, …, n successes, estimate p = mean/n, then work out the expected frequencies N·C(n, x)pˣqⁿ⁻ˣ and compare them with the observed ones (with the chi-square statistic).
The formula
P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ
- Mean
- np
- Variance
- np(1 − p)
Worked example
10 coin flips, exactly 5 heads
- C(10, 5) × 0.5¹⁰ = 252 ÷ 1024
- = 0.2461
Questions people ask
When can I use the normal approximation?
When np and n(1 − p) are both at least about 10. This calculator computes the exact value anyway.
What makes trials independent?
One result doesn't change the chance of another, like separate coin flips.