Binomial Distribution

PMF and CDF for n independent trials

X ~ B(n = 10, p = 0.5) · P(X = 5)

P(X = k) — exactly k

0.246094

P(X = k)

0.246094

P(X ≤ k)

0.623047

P(X ≥ k)

0.623047

Mean (μ = np)

Variance (σ²)

2.5

Std. deviation (σ)

1.5811

About this calculator

The Binomial Distribution Calculator finds the probability of k successes in n independent trials, each succeeding with probability p. It plots the full probability mass function (PMF) and reports the exact probability P(X = k), the cumulative P(X ≤ k), the upper tail P(X ≥ k), and the range P(a ≤ X ≤ b), along with the mean np, variance np(1−p), and standard deviation. Useful for coin flips, quality control, polling, A/B testing, and probability homework.

Common examples

10 fair coin flips, exactly 5 heads: P(X = 5) = 0.2461
10 fair coin flips, at most 5 heads: P(X ≤ 5) = 0.6230
20 trials at p = 0.3, at least 10 successes: P(X ≥ 10) = 0.0480
Between 4 and 6 heads in 10 fair flips: P(4 ≤ X ≤ 6) = 0.6563
n = 100, p = 0.5: mean = 50, σ = 5