Binomial Distribution Calculator
Enter the number of trials, successes, and probability per trial below to instantly get the exact probability, mean, and variance.
P(X = k) for n independent trials, each with success probability p
How Binomial Probability Is Calculated
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), where C(n, k) is the number of ways to choose which k of the n trials are successes.
Worked example
10 coin flips (n = 10), probability of exactly 3 heads (k = 3), p = 0.5.
C(10, 3) = 120
P(X = 3) = 120 × 0.5³ × 0.5⁷ = 120 × 0.125 × 0.0078125
P(X = 3) ≈ 0.1172 (about 11.7%)
Frequently Asked Questions
What is a binomial distribution?
A binomial distribution describes the number of successes in a fixed number of independent trials, where each trial has the same probability of success — like the number of heads in 10 coin flips, or the number of defective parts in a batch of 50.
What conditions does a binomial distribution require?
A fixed number of trials, each trial independent of the others, only two possible outcomes per trial (success or failure), and the same success probability on every trial. If any of these don't hold, the binomial formula doesn't apply.
What's the difference between P(X = k) and P(X ≤ k)?
P(X = k) is the probability of exactly k successes. P(X ≤ k), the cumulative probability, is the probability of k or fewer successes — the sum of P(X = 0) through P(X = k). This calculator gives the exact P(X = k) value.
How do you find the mean and variance of a binomial distribution?
Mean = n × p (the number of trials times the success probability). Variance = n × p × (1 − p). These come directly from the distribution's definition, without needing to compute individual probabilities.
Related Calculators
- Standard Deviation Calculator — Statistics
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