Binomial Probability CalculatorFree online math calculator — instant results, no sign-up

P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), where C(n, k) = n! / [k!(n − k)!]. For the default inputs: P(X = 3) = [10! / (3! × 7!)] × 0.4^3 × 0.6^7 = 120 × 0.064 × 0.0279936 = 0.214990848.

• XThe random variable counting successes across all trials.
• nThe total number of trials; 10 in the worked example.
• kThe exact number of successes of interest; 3 in the worked example.
• pThe success probability on each trial; 0.4 in the worked example.
• 1 − pThe failure probability on each trial; 0.6 in the worked example.
• C(n, k)The number of ways to place k successes among n trials; C(10, 3) = 120.
• !Factorial notation: the product of positive integers up to the given integer, with 0! = 1.
• P(X = k)The probability of exactly k successes, regardless of their order.

Check that n and k are whole numbers with 0 ≤ k ≤ n and that 0 ≤ p ≤ 1. Calculate the number of possible success arrangements using C(n, k). Multiply this count by p^k, the success contribution, and (1 − p)^(n − k), the failure contribution. For n = 10, k = 3, and p = 0.4, this gives 0.214990848.

These are the calculator's starting values and the results it actually produces for them.

1. 1Enter 10 in Trials to specify the total number of trials.
2. 2Enter 3 in Successes to request exactly 3 successes.
3. 3Enter 0.4 in Success probability, using a decimal rather than a percentage.
4. 4Run the calculation to evaluate C(10, 3) × 0.4^3 × 0.6^7.
5. 5Read the Result of 0.214990848 as the probability of exactly 3 successes in 10 trials.

Further reading

• Khan Academy