Binomial Probability Calculator
Exactly k, at most k, at least k — plus the mean.
Work out Binomial Probability. Exactly k, at most k, at least k — plus the mean. Names the mistake before you make it.
Probability of exactly 5 successes
24.6094%
mean 5 · sd 1.581
P(exactly k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ. The combination counts how many orderings give k successes, and the powers give the probability of any one of them. It needs a fixed number of independent trials each with the same probability. Drawing cards without replacement breaks that — the probability shifts as the deck depletes — and needs the hypergeometric distribution instead. The mean is np, which is worth checking against intuition: 10 flips of a fair coin should average 5 heads, and any answer far from that suggests an input is wrong.
How the Binomial Probability Calculator works
Enter trials, successes and the per-trial probability for the exact probability and both cumulative tails. The mean is np, which is worth checking against intuition — an answer far from it usually means an input is wrong.
Also known as: binomial distribution calculator · probability of exactly k successes · coin flip probability calculator · n choose k probability
Frequently asked questions
What is the binomial formula?
P(k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ. The combination counts how many orderings produce k successes; the powers give the probability of any single one of them.
When does the binomial distribution apply?
Fixed number of trials, each independent, each with the same probability, each a success or a failure. Drawing cards without replacement breaks the third condition and needs the hypergeometric distribution instead.
What is the probability of exactly 5 heads in 10 flips?
About 24.6%, or 252/1024. Lower than most people expect — the single likeliest outcome still happens less than a quarter of the time, because there are so many other results available.
What are the mean and standard deviation?
np and √(np(1−p)). For 10 fair flips that is 5 and about 1.58, so most results land between 3 and 7.
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