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Binomial Distribution Calculator

The binomial distribution describes the number of successes in n independent trials, each with success probability p. Enter n, p and a target k to get the point probability P(X = k), the cumulative probabilities and the distribution's mean, variance and standard deviation.

Binomial Distribution Calculator

P(X = k), P(X ≤ k), P(X ≥ k) plus mean and variance for Binomial(n, p).

Try:
AnswerP(X = 5) = 0.246094, P(X ≤ 5) = 0.623047, P(X ≥ 5) = 0.623047
  1. DistributionX ~ Binomial(n = 10, p = 0.5)
  2. P(X = k)C(10, 5)·0.5^5·0.5^5 = 0.246094
  3. P(X ≤ k)0.623047
  4. P(X < k)0.376953
  5. P(X ≥ k)1 − P(X < k) = 0.623047
  6. P(X > k)0.376953
  7. Meanμ = n·p = 5
  8. Varianceσ² = n·p·(1 − p) = 2.5
  9. Standard deviationσ = 1.58114
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Worked examples

Binomial probability: P(X = 5) for n = 10, p = 0.5 Compute a binomial point probability and the cumulative tails.

Frequently asked questions

When is the binomial distribution appropriate?

When the trials are independent, each has exactly two outcomes (success / failure), and the success probability p stays constant across trials.

What is the mean of a binomial?

μ = n·p — the average number of successes you would expect across many repetitions of the n-trial experiment.

How is P(X ≤ k) computed?

By summing the point probabilities P(X = 0), P(X = 1), …, P(X = k). The calculator does this internally with log-arithmetic to stay accurate for large n.

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