Binomial Distribution Calculator
A binomial distribution calculator finds the chance of exactly k successes in n repeated yes-or-no trials with the same success probability p. Enter n, p and k to get P(X = k), at most, at least, the mean and standard deviation, with every step of the working shown.
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Working
Math Lantern Team · Updated
How the binomial distribution calculator works
Use the binomial distribution when you repeat the same yes-or-no trial a fixed number of times and each trial has the same chance of success. Enter n, p and k to get the chance of exactly k successes, at most k, at least k, plus the mean and standard deviation, all with the working shown.
Worked example: 3 heads in 10 flips
For a fair coin, n = 10, p = 0.5 and k = 3. There are C(10, 3) = 120 ways to place three heads among ten flips. Each particular arrangement has probability 0.5³ × 0.5⁷ = 0.0009765625. Multiply: 120 × 0.0009765625 = 0.1171875, so there is an 11.72 % chance of exactly three heads. Adding the probabilities for 0, 1, 2 and 3 heads gives P(X ≤ 3) = 0.171875.
When the binomial model fits
- A fixed number of trials, n.
- Only two outcomes per trial, success or failure.
- The same probability p on every trial.
- Trials independent of each other.
Drawing cards without replacement or surveying a small group without replacement breaks independence; use the hypergeometric distribution instead.
Mean and standard deviation
The expected number of successes is n × p. The spread is √(n × p × (1 − p)). For 10 fair flips that is a mean of 5 and a standard deviation of 1.58.
The calculator uses double-precision arithmetic in your browser. On 12 October 2026 it matched exact integer arithmetic on 140 random cases with n up to 1,000, the largest difference being 0.0000005. Results are rounded to six decimals, so very tiny probabilities are shown as 0.
Frequently asked questions
What is the difference between P(X = k) and P(X ≤ k)?
P(X = k) is the chance of exactly k successes. P(X ≤ k) adds the chances of 0, 1, up to k successes.
How do I find “at least k” successes?
P(X ≥ k) = 1 − P(X ≤ k − 1). The table shows it directly.
Can p be a percentage or a fraction?
Yes. You can type 0.25, 1/4 or 25%.
What if my trials are not independent?
Then the binomial formula is not appropriate. Use a model that accounts for dependence, such as the hypergeometric distribution.
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