Binomial Probability Calculator

Updated 4 November 2026 · Standard formula, shown below

Probability of exactly k successes 0.117188

Expected number of successes: 5.00 successes; Result
Standard deviation: 1.5811 successes; Result
Number of trials n: 10 trials; You entered

This calculator gives the probability of getting exactly k successes in n independent trials when each trial succeeds with the same probability p. It also returns the expected number of successes and the standard deviation of the binomial distribution.

Formula

P(X = k) = C(n, k) × p^k × (1 − p)^{n − k}, where C(n, k) = n! ÷ (k! (n − k)!). Mean μ = n p, standard deviation σ = √(n p (1 − p)).

n: number of trials (whole number)
k: number of successes, 0 to n
p: probability of success on each trial, 0 to 1 (enter 0.25, not 25)

Worked Example

n = 10 coin tosses, k = 3 heads, p = 0.5.

C(10, 3) = 120
0.5³ = 0.125 and 0.5⁷ = 0.0078125
P = 120 × 0.125 × 0.0078125 = 0.117188 (about 11.7%)
μ = 10 × 0.5 = 5.00, σ = √(10 × 0.5 × 0.5) = √2.5 = 1.5811

Limits and common mistakes

The trials must be independent and p must stay the same on every trial. Drawing cards without replacement is not binomial (use the hypergeometric distribution).
This gives “exactly k”. For “at most k” add the probabilities for 0, 1, … k.
Enter p as a decimal fraction, not a percentage.

Formula used

in n | Number of trials n | trials | 10 | 0
in k | Number of successes k | successes | 3 | 0
in p | Probability of success on one trial | - | 0.5 | 0
out prob | Probability of exactly k successes | - | ncr(n, k) * p^k * (1 - p)^(n - k) | 6
out mu | Expected number of successes | successes | n * p | 2
out sdev | Standard deviation | successes | sqrt(n * p * (1 - p)) | 4

Each line is one input (in, sel, list) or one result (out: label, unit, formula, decimals). How the calculators work

Method: the standard formula for this calculation, rounded as each result states. See how the calculators work or report an error.