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)
5
Variance (σ²)
2.5
Std. deviation (σ)
1.5811
Bar chart of the binomial PMF with the selected outcomes highlighted. P(X = 5) = 0.246094Bar chart of the binomial PMF with the selected outcomes highlighted. P(X = 5) = 0.246094012345678910

この計算機について

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.

一般的な例

  • 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