Calculate binomial probability, the probability of exactly k successes (and the cumulative probability of k or fewer) across n independent trials, each with the same probability of success.
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For n independent trials, each with the same probability p of success, the probability of exactly k successes is P(X=k) = C(n,k) × p^k × (1−p)^(n−k), where C(n,k) is the combination 'n choose k'. This calculator also sums P(X=i) for every i from 0 to k to give the cumulative probability P(X≤k), the chance of getting k or fewer successes.
The binomial distribution applies when you have a fixed number of independent trials, each with only two possible outcomes (success or failure), and the same probability of success on every trial, classic examples include counting heads in a series of coin flips, or counting defective items in a batch where each item independently has the same defect probability.
For n = 10 coin flips, p = 50% chance of heads per flip, k = 6: the exact probability of getting precisely 6 heads is P(X=6) = C(10,6) × 0.5⁶ × 0.5⁴ ≈ 20.51%. The cumulative probability of getting 6 or fewer heads, P(X≤6), sums the exact probabilities for 0 through 6 heads, giving about 82.81%. The expected (mean) number of heads is n × p = 10 × 0.5 = 5, and the standard deviation is √(n×p×(1−p)) = √(10×0.5×0.5) = √2.5 ≈ 1.58.
Confusing exact probability P(X=k) with cumulative probability P(X≤k), the exact probability of precisely 6 heads out of 10 flips is about 20.5%, but the probability of 6 or fewer heads is much higher, about 82.8%, since it includes every outcome from 0 through 6.
Using the binomial distribution for trials that aren't actually independent or don't share the same success probability, like drawing cards from a deck without replacement, that changes the probability on each draw and needs a different (hypergeometric) model.
Entering the probability as a decimal (0.5) instead of a percentage (50), this calculator's probability field expects a percentage between 0 and 100.
Exact probability, P(X=k), is the chance of getting precisely k successes. Cumulative probability, P(X≤k), is the chance of getting k or fewer successes, the sum of the exact probabilities for every outcome from 0 up to k.
The mean (n × p) is the expected number of successes on average across many repetitions. The standard deviation measures how much the actual number of successes typically varies from that average, a larger standard deviation means more variability from trial set to trial set.
No, the binomial distribution specifically assumes the same success probability on every trial. If probabilities vary trial to trial, a different (non-binomial) model is needed.
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