Binomial Probability Calculator

The binomial distribution counts the number of successes in n independent trials, where each trial has the same probability p of success and only two possible outcomes (success or failure). The binomial probability P(X = k) is the chance of getting exactly k successes in those n trials.

P(X = k) = C(n, k) × p^k × (1 − p)^(n − k). C(n, k) is the number of ways to choose which k of the n trials are successes (the binomial coefficient). p^k is the probability that those specific trials are all successes. (1 − p)^(n − k) is the probability that the other trials are all failures. Multiplying combines them into the probability of any single arrangement plus the count of arrangements.

The binomial model requires four conditions. (1) A fixed number of trials n. (2) Each trial is independent of the others. (3) Each trial has the same probability p of success. (4) Each trial has exactly two outcomes (success or failure). Sampling without replacement from a small population violates independence; flipping a fair coin many times satisfies all four conditions.

P(X ≤ k) is the probability of getting k or fewer successes (the cumulative probability up through k). P(X ≥ k) is the probability of getting k or more successes (the upper-tail cumulative). They are complementary: P(X ≥ k) = 1 − P(X ≤ k − 1). Use 'at most' when asking 'no more than this many', 'at least' when asking 'this many or more', and 'exactly' for a single specific count.

The mean (expected value) of a binomial distribution is n × p. With 100 trials at p = 0.3, you expect 30 successes on average. The variance is n × p × (1 − p), and the standard deviation is the square root of the variance. These help you judge whether a specific result is unusual.

For n above about 30, the normal distribution is often a reasonable approximation, especially when both np and n(1 − p) are at least 5. Use the normal-distribution calculator with mean = np and standard deviation = √(np(1 − p)). For exact binomial probabilities on large n, statistical software handles arbitrary sizes; this calculator caps n at 1,000 to keep the in-browser loop responsive.