Binomial Distribution Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/5/2026
Binomial probability is calculated with P(X=k) = C(n,k) × p^k × (1−p)^(n−k). For 10 flips of a fair coin, the probability of getting exactly 3 heads is about 11.72%.
Explanation
The binomial distribution models the number of successes obtained over a fixed number of independent trials, where each trial has only two possible outcomes (success or failure) with the same probability of success p every time — a pattern called a Bernoulli trial repeated n times. The formula combines three elements: the number of different ways to get exactly k successes out of n trials (C(n,k), the number of combinations, calculated with our combinations and permutations calculator), the probability that k specific trials succeed (p^k), and the probability that the remaining n−k trials fail ((1−p)^(n−k)). This model applies to many real situations: the probability of getting a given number of heads over several coin flips, the number of defective parts in a production batch with a known defect rate, or the number of favorable respondents in a survey with a known approval rate — as long as each trial is genuinely independent of the others and the probability of success stays constant from one trial to the next.
Example: 10 coin flips, probability of 3 heads
Inputs
Number of trials: 10. Successes sought: 3. Probability of success: 50%.
Calculation
P(X=3) = C(10,3) × 0.5³ × 0.5⁷ = 120 × 0.125 × 0.0078125 ≈ 0.1172, or 11.72%.
Result
The probability of getting exactly 3 heads in 10 flips is about 11.72%.
Frequently asked questions
What's the difference from a simple combination C(n,k)?
The combination C(n,k) only counts the number of ways to get k successes out of n trials, without accounting for their individual probability. The binomial distribution goes further by weighting this count by the actual probability of that specific scenario occurring, giving a real probability (between 0 and 100%) rather than a plain count of possibilities.
Does this formula apply to any situation with two possible outcomes?
Only if two conditions are met: each trial must be independent of the others (the outcome of one trial doesn't influence the next), and the probability of success must stay constant from one trial to the next. Sampling without replacement from a finite population (where each draw changes the remaining probabilities) doesn't exactly follow a binomial distribution, but a hypergeometric one.
How do I calculate the probability of at least k successes, rather than exactly k?
You need to add up the probabilities for k, k+1, k+2, up to n successes (or subtract from 100% the sum of the probabilities for 0 through k−1 successes). This calculator gives the probability of an exact number of successes; for a cumulative probability, repeat the calculation for each relevant value of k and add up the results.