Probability · Distributions
Binomial Distribution Calculator
The binomial distribution counts successes in a fixed number of independent yes/no trials — coin flips, defective parts, free throws made. Set the trials n, the success probability p and a count k, and get every probability at once: exactly k, fewer than, at most, at least and more than — with the full distribution drawn, the mean and standard deviation, and an honest note on when the tidy normal-curve shortcut is allowed to stand in.
p is the chance of a success on a single trial; k is how many successes you are asking about. All five probabilities are listed below — the chip just chooses which one to headline and shade.
Result
In plain English
Flip a biased coin n times and count the heads. The binomial distribution gives the probability of every possible count, from 0 to n. Each individual flip is a single yes/no trial (a Bernoulli trial); the binomial is just n of them added up. Two numbers fix everything: how many trials, n, and the chance of success on each one, p. From those you get the whole shape — its centre at n·p, and a spread that is widest when p is near ½.
- binomial distribution
- The probability distribution of the number of successes in n independent trials, each succeeding with the same probability p. Written X ~ B(n, p).
- P(X = k), the probability mass
- The chance of exactly k successes: C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ. The C(n, k) counts the orderings; the powers give the probability of any one such ordering.
- cumulative: at most / at least
- P(X ≤ k) adds up the bars from 0 to k; P(X ≥ k) adds from k to n. These “or more / or fewer” questions are usually what you actually want, not the exact-k figure.
- mean and standard deviation
- The expected number of successes is μ = n·p; the spread is σ = √(n·p·(1 − p)). On average you get n·p, give or take about σ.
- the four assumptions (BINS)
- Binary outcomes, Independent trials, a fixed Number of them, and the Same probability p throughout. Break any one — sampling without replacement, a streak that changes the odds — and the binomial no longer applies.
- the normal approximation
- When n·p and n·(1 − p) are both at least about 10, the binomial is close to a normal curve with the same mean and SD — convenient, but only a shortcut, and a poor one in the tails or for small n.
Frequently asked
How do you calculate a binomial probability?
The probability of exactly k successes in n trials, each with success probability p, is P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ, where C(n, k) = n! ∕ [k!(n − k)!] is the number of ways to choose which k of the n trials succeed. For “at most k” or “at least k,” add the relevant individual probabilities. This calculator does all of that and lists exactly-k, fewer-than, at-most, at-least and more-than side by side.
What is the difference between “exactly,” “at least” and “at most”?
“Exactly k” is a single bar of the distribution, P(X = k). “At most k” (P(X ≤ k)) sums every bar from 0 up to and including k; “at least k” (P(X ≥ k)) sums from k up to n. Watch the endpoint: “more than k” excludes k itself (P(X > k) = P(X ≥ k + 1)), and “fewer than k” excludes it too. The exact-k probability is often surprisingly small even at the peak, which is why real questions almost always ask for a cumulative range.
When can I use the normal approximation to the binomial?
The usual rule of thumb is that both n·p and n·(1 − p) should be at least about 10 (some texts say 5). Then the binomial is roughly normal with mean n·p and standard deviation √(n·p·(1 − p)), and you can use z-scores. It is only an approximation, though: it is weakest in the tails and when p is close to 0 or 1, and a continuity correction (adding or subtracting ½) helps. With a small n, skip it and use the exact binomial — which is what this page computes.
What is the difference between binomial and Bernoulli?
A Bernoulli trial is a single yes/no experiment with success probability p — one coin flip. The binomial distribution is the number of successes across n independent Bernoulli trials with the same p. So Bernoulli is the special case n = 1, and a binomial variable is a sum of n independent Bernoulli variables. If the number of trials is not fixed in advance, or p changes, or trials are dependent, you need a different model (geometric, negative binomial, hypergeometric, or Poisson).