Binomial CDF Calculator — P(X ≤ k) for B(n, p)

Compute P(X ≤ k), P(X = k) and P(X ≥ k) for a binomial distribution B(n, p) — exact values, no approximation.

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Formula

P(X ≤ k) = Σ C(n,i)·p^i·(1−p)^(n−i) for i = 0..k, computed with log-factorials so n up to 5,000 stays numerically stable. Exact summation — not the normal approximation.

Examples

InputResult
n=20, p=0.3, k=5P(X≤5)=0.416371 · P(X=5)=0.178863
n=10, p=0.5, k=7P(X≤7)=0.945313
n=100, p=0.05, k=3P(X≤3)=0.257839

Frequently Asked Questions

What is a binomial distribution?

The count of successes in n independent yes/no trials with the same success probability p — coin flips, quality-control defect counts, free throws made out of attempts.

When is the normal approximation acceptable?

A common rule is np ≥ 5 and n(1−p) ≥ 5. This calculator always uses exact summation, so the answer stays correct even for small n or extreme p where the approximation fails.

What are the mean and variance?

Mean = np, variance = np(1−p). For B(20, 0.3) the mean is 6 and the standard deviation is √4.2 ≈ 2.05.

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