Binomial Probability Calculator
Enter the number of trials, the number of successes you want, and the per-trial success probability to get the exact chance — the count behind coin flips, free throws, and quality checks.
Three values, one probability
Enter the trials (n), the successes (k), and the per-trial probability (p); the calculator returns P(X = k) = C(n, k) · pᵏ · (1 − p)^(n − k).
Exactly k successes
The result is the chance of exactly k successes — not at least k. For ranges, add the individual probabilities together.
What is binomial probability?
The chance of exactly k successes in n trials
The binomial probability calculator finds the chance of getting exactly k successes in n independent trials, where every trial has the same success probability p. Each trial is a yes/no experiment — a coin lands heads or tails, a free throw goes in or misses, a part passes or fails — and the binomial formula counts how the successes can fall. It answers questions like "what is the chance of exactly 5 heads in 10 coin flips" with a single number between 0 and 1, which you can also read as a percentage.
Enter your trials, successes, and per-trial probability to get the exact binomial probability instantly.
One formula combines the number of ways the successes can occur with how likely each arrangement is.
P(X = k) = C(n, k) · pᵏ · (1 − p)^(n − k)The binomial coefficient C(n, k) counts how many different orderings give exactly k successes. The term pᵏ is the chance that those k successes all happen, and (1 − p)^(n − k) is the chance that the remaining n − k trials all fail. Multiplying the three together gives the probability of exactly k successes in any order. Use a probability p between 0 and 1, and whole numbers for n and k with k no greater than n.
Suppose you flip a fair coin 10 times and want the chance of exactly 5 heads.
Count the orderings
C(10, 5) = 252 — the number of ways 5 heads can fall among 10 flips.
Multiply the trial probabilities
0.5⁵ × 0.5⁵ = 0.5¹⁰ = 0.000977 — the chance of one specific sequence.
Combine
252 × 0.000977 = 0.246094, or about 24.6% — the chance of exactly 5 heads.
The number you get is the probability of exactly your chosen count of successes, expressed both as a decimal between 0 and 1 and as a percentage. Notice that this is a single slice of the whole distribution: if you summed the probability over every possible count from 0 to n, the total would be exactly 1, because one of those counts has to happen. That also explains why the most likely single outcome — near n × p successes — rarely tops 25% even for a fair coin; the probability is spread across many possible counts. To answer "at least k" or "at most k", add the individual P(X = k) values across the range you care about. And remember the result only holds when the trials are genuinely independent with a constant p — a streak of made free throws that changes a player's confidence would break that assumption.
The binomial formula is exact, but it rests on assumptions that must actually hold.
Fixed trials, constant p, independence, k ≤ n
This calculator assumes a fixed number of independent trials, two outcomes per trial, and the same success probability p on every trial. If trials affect one another, or p drifts between trials, the binomial model no longer fits — and sampling without replacement from a small group needs the hypergeometric distribution instead. Both n and k must be whole numbers with k no greater than n, and p must lie between 0 and 1, otherwise the calculator returns no result.