Binomial Distribution Calculator
Enter the number of trials, the chance of success on each trial and the number of successes k.
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Formula
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k)P(X ≤ k)is the sum ofP(X = i)for i from 0 to kP(X ≥ k) = 1 − P(X ≤ k − 1)- Mean
n × p, variancen × p × (1 − p)
Worked example
n = 20 trials, p = 0.3, k = 6:
- P(X = 6) = 0.1916
- P(X ≤ 6) = 0.6080
- P(X ≥ 6) = 0.5836
- Mean = 20 × 0.3 = 6, variance = 20 × 0.3 × 0.7 = 4.2
Frequently asked questions
What is the binomial probability formula?
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k). For 10 fair coin flips and exactly 5 heads, C(10, 5) = 252, so P = 252 ÷ 1024 = 0.2461.
What is the difference between P(X = k), P(X ≤ k) and P(X ≥ k)?
P(X = k) is exactly k successes. P(X ≤ k) adds up 0 through k. P(X ≥ k) adds up k through n. With n = 20 and p = 0.3, k = 6 gives 0.1916, 0.6080 and 0.5836.
What are the mean and variance of a binomial distribution?
The mean is n × p and the variance is n × p × (1 − p). For n = 20 and p = 0.3 the mean is 6 and the variance is 4.2.
When can I use the binomial distribution?
When there is a fixed number of independent trials, each with only two outcomes and the same success probability. Drawing without replacement from a small group breaks independence.