Binomial distribution calculator with workings
Enter n, p and k. You get the probability of exactly, at most, at least, fewer than and more than k successes, plus mean, variance and standard deviation.
Result
- P(X = 3)
- 0.266828
- P(X ≤ 3)
- 0.649611
- P(X ≥ 3)
- 0.617217
- P(X < 3)
- 0.382783
- P(X > 3)
- 0.350389
- Expected value E(X) = np
- 3
- Variance np(1 − p)
- 2.1
- Standard deviation
- 1.4491
Workings
- 1Binomial coefficient: C(10, 3) = 10! / (3! × 7!) = 120
- 2P(X = 3) = (10 choose 3) · 0.3^3 · 0.7^7 = 0.266828
- 3P(X ≤ 3) = P(X = 0) + … + P(X = 3) = 0.649611
- 4P(X ≥ 3) = 1 − P(X ≤ 2) = 0.617217
- 5E(X) = 10 × 0.3 = 3; Var(X) = 10 × 0.3 × 0.7 = 2.1
The formula
P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ
“At least k” is 1 − P(X ≤ k − 1): k itself is included.
Frequently asked questions
What does C(n, k) mean?+
The number of ways to place k successes among n trials: n! / (k!(n − k)!).