Calculate exact and cumulative binomial probabilities, plus distribution table and chart.
The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success.
Classic examples:
P(X = k): The probability of getting exactly k successes in n trials. For example, P(X=6) when flipping 10 fair coins is the chance of getting exactly 6 heads.
P(X ≤ k): Cumulative probability — the chance of getting k or fewer successes. This is the CDF at k.
P(X ≥ k): The chance of getting k or more successes. This is 1 − P(X ≤ k−1). Useful for asking "what is the probability of at least k successes?"
If sampling without replacement from a small population, use the hypergeometric distribution instead.
When n is large and p is not too close to 0 or 1, the binomial distribution is well approximated by the normal distribution with μ = np and σ = √(np(1−p)).
A common rule: the normal approximation is adequate when np ≥ 5 and n(1−p) ≥ 5. Below these thresholds, use the exact binomial probability from this calculator.