Binomial probability calculator

Compute exact-count and tail probabilities for independent fixed-probability binary trials, retaining logarithms for very small probabilities.

Your input

Outputs are cleared when the input changes.

Result

Enter your data or use the example, then run the tool.

Worked example

n=10, p=0.5, k=3: exactly 0.1171875, at most 0.171875, at least 0.9453125.

With n=0, the count is certainly zero. With p=1, the count is certainly n.

Method

Use C(n,k)p^k(1−p)^(n−k). Log factorials and log1p form terms without overflowing factorials.

Selected terms are summed directly with log-sum-exp, including upper tails; p=0, p=1 and n=0 are handled separately.

Limits

n must be an integer 0–10,000, counts 0–n, and p 0–1. Trials must be independent, binary and share fixed p.

Floating-point approximation; an underflowed probability is labelled below range, not impossible. No medical, gambling or outcome prediction.

Sources

NIST: binomial distribution

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