Probability Distributions Calculator
Binomial, Poisson and normal probabilities — exact and cumulative — with the standardisation and the sums shown. The same functions that sit behind statistics tables, computed in the open.
Example — Exactly 5 heads in 10 fair coin flipsP(X = 5) = 24.6094%
How we got it — the example, step by step
- Binomial with n = 10, p = 0.5: P(X = 5) = C(10, 5) × 0.5⁵ × 0.5⁵.
- C(10, 5) = 252, and 0.5¹⁰ = 1/1024, so P = 252/1024 = 24.6094%.
Check it backwards
P(X ≤ 5) comes out 62.3047% — past halfway, as it must be, because the distribution is symmetric and P(X = 5) itself straddles the middle.
Questions people also ask
How does the normal calculator get Φ without a table?
From the error function, using the Abramowitz–Stegun approximation (accurate to about 7.5 × 10⁻⁸). The sanity anchors still hold exactly as they should: Φ(0) = 50%, Φ(1.96) ≈ 97.50%, and ±1σ ≈ 68.27%.