Binomial Probability Calculator

Binomial probability

The binomial probability tells you the chance of getting exactly k successes across n independent trials, where each trial succeeds with probability p: P(X = k) = C(n,k) · p^k · (1−p)^(n−k). Here C(n,k) is the binomial coefficient. Say you flip a fair coin 10 times. The odds of landing exactly 7 heads work out to C(10,7) · 0.5⁷ · 0.5³ ≈ 0.117, or 11.7%. Once both n·p > 5 and n·(1−p) > 5 hold, a Normal(np, np(1−p)) curve approximates the binomial quite well, which follows from the Central Limit Theorem.

Applications

It shows up in A/B testing of conversion rates, in quality control when you count defective items in a batch, and in Mendelian genetics for predicting phenotype proportions in offspring. Reliability engineering uses it for k failures out of n components, and you'll also find it behind opinion polls and clinical trial response rates.

FAQ

When does the binomial not apply? Two cases break it. If the trials depend on each other (think small populations sampled without replacement, where the hypergeometric fits instead), or if p shifts from one trial to the next.

P(X = k) or P(X ≤ k)? P(X = k) is the probability of hitting that exact count. If instead you want "at most k successes", you need the cumulative P(X ≤ k), which adds up P(X = i) for i = 0..k.

When to use the normal approximation? Reach for it once both np > 5 and n(1−p) > 5. Below that threshold, stick with the exact binomial formula.