Binomial Distribution Calculator

The binomial distribution counts the successes in a fixed number of tries when each try has only two outcomes (success or failure), the same chance of success every time, and the tries don’t affect each other. Coin flips, free throws, faulty parts in a batch, people who answer “yes” in a survey and passengers who don’t turn up for a flight all follow it.

The formula

P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ
C(n, k) = n! ÷ (k! × (n − k)!), the ways to pick which k tries succeed
P(X ≤ k) = P(X = 0) + P(X = 1) + … + P(X = k)
P(X < k) = P(X ≤ k − 1)
P(X ≥ k) = 1 − P(X < k)
P(X > k) = 1 − P(X ≤ k)
mean = n × p, variance = n × p × (1 − p), SD = √variance
at least one success = 1 − (1 − p)ⁿ
normal approximation: P(X ≤ k) ≈ Φ((k + 0.5 − np) ÷ SD)
Poisson approximation: P(X = k) ≈ e^(−λ) × λᵏ ÷ k!, with λ = n × p

Worked example

Flip a fair coin 10 times (n = 10, p = 0.5). What is the chance of exactly 5 heads?

The number of ways to pick which 5 of the 10 flips are heads is C(10, 5) = 10! ÷ (5! × 5!) = 3,628,800 ÷ (120 × 120) = 3,628,800 ÷ 14,400 = 252. Each of those ways has chance 0.5⁵ × 0.5⁵ = 0.5¹⁰ = 1/1024 = 0.0009765625. So P(X = 5) = 252 × 0.0009765625 = 252/1024 = 0.24609375, about 24.61%.

For at most 5 heads, add the bars from 0 to 5. The ways are C(10, 0) to C(10, 5) = 1 + 10 + 45 + 120 + 210 + 252 = 638, so P(X ≤ 5) = 638/1024 = 0.623046875. Fewer than 5 heads leaves out the 252: P(X < 5) = 386/1024 = 0.376953125. Because a fair coin is symmetric, P(X ≥ 5) = 0.623046875 and P(X > 5) = 0.376953125 too.

The mean is 10 × 0.5 = 5 heads, the variance is 10 × 0.5 × 0.5 = 2.5 and the SD is √2.5 = 1.58114. The chance of at least one head is 1 − 0.5¹⁰ = 1 − 1/1024 = 1023/1024 = 0.9990234375.

The normal approximation with a continuity correction treats “exactly 5” as the strip from 4.5 to 5.5: z = (5.5 − 5) ÷ 1.58114 = 0.31623, so P(X = 5) ≈ Φ(0.31623) − Φ(−0.31623) = 0.62409 − 0.37591 = 0.24817, close to the exact 0.24609 even though np = 5 is below the usual rule of 10. The Poisson approximation with λ = 5 gives e⁻⁵ × 5⁵ ÷ 5! = 0.17547, which is far off, because p = 0.5 is not small.

Which approximation, and when

Reading the words in a question

• Exactly k: P(X = k). At most k or no more than k: P(X ≤ k).
• Fewer than k or less than k: P(X < k). At least k or k or more: P(X ≥ k).
• More than k: P(X > k). Between a and b (inclusive): P(a ≤ X ≤ b), in Advanced.

Spreadsheets and notes

• Excel and Google Sheets give P(X = k) with BINOM.DIST(k, n, p, FALSE), P(X ≤ k) with BINOM.DIST(k, n, p, TRUE), a range with BINOM.DIST.RANGE(n, p, a, b) and the inverse with BINOM.INV(n, p, target).
• Each P(X = k) is worked out with logarithms of factorials (ln n!), then added from the short end, so tiny tails such as 7.56 × 10⁻³¹ keep their accuracy. For n up to 170 the exact C(n, k) is used, and the working shows it in full when it has 40 digits or fewer.
• When p is a simple fraction or decimal and n is 200 or less, the fraction view gives the exact answer, like 63/256.
• If the chance changes from try to try (cards drawn without putting them back), the binomial is only an approximation. For dice, cards and Bayes’ theorem, use the probability calculator; for C(n, k) on its own, the permutation and combination calculator; and for areas under a bell curve, the z-score calculator.