Binomial Distribution Calculator — probability of k successes in n trials

How it works

The binomial distribution answers a very common question: if you repeat the same yes/no trial n times, each succeeding with probability p, what's the chance of exactly k successes? Three pieces multiply together:

P(X=k) = C(n,k) × p^k × (1 − p)^(n − k)

• p^k — the probability that k particular trials all succeed;
• (1 − p)^(n − k) — the probability the other n − k all fail;
• C(n,k) — the number of different ways those k successes can be arranged among the n trials (the combination, "n choose k").

The combination is essential: there are many patterns of which trials succeed, and each has the same probability, so you multiply by how many there are. The trials must be independent and share the same p.

Worked example

What's the probability of exactly 3 heads in 10 fair coin flips? Here n = 10, k = 3, p = 0.5:

P = C(10,3) × 0.5³ × 0.5⁷ = 120 × 0.5¹⁰ = 120 ÷ 1024 ≈ 0.117 (11.7%)

The 120 counts the ways to choose which 3 of the 10 flips are heads, and 0.5¹⁰ is the probability of any one specific pattern. Change the odds and it shifts: getting 5 of 20 questions right by guessing with p = 0.25 is about 20.2%, while all 5 of 5 at p = 0.5 is just 0.5⁵ = 3.125%.

Frequently asked questions

What is the binomial distribution used for?

Any situation with a fixed number of independent yes/no trials at a constant success rate: coin flips, quality-control pass/fail, multiple-choice guessing, conversion rates, free-throw makes. It tells you how likely each possible number of successes is.

What are n, k and p?

**n** is the number of trials, **k** the number of successes you're asking about, and **p** the probability of success on a single trial. The formula needs all three, and k can be any whole number from 0 to n.

What conditions does it assume?

Four: a **fixed** number of trials n; only **two outcomes** per trial (success or failure); the same **success probability p** every trial; and trials that are **independent** (one result doesn't affect another). If p changes between trials, or trials influence each other, the binomial model doesn't apply.

How do I find "at least k" or "at most k" instead of exactly k?

Add up the individual probabilities. "At most k" is P(0) + P(1) + ... + P(k); "at least k" is 1 minus the probabilities below k. This calculator gives the exact-k probability P(X=k); sum several of those for a range.

Why multiply by the combination C(n,k)?

Because there are many different orders in which k successes can happen among n trials, and each order has the same probability p^k(1−p)^(n−k). C(n,k) counts those orders, so multiplying by it accounts for every arrangement rather than just one.

What's the average number of successes?

For a binomial distribution the mean is n × p — so 10 flips of a fair coin average 5 heads, and 20 guesses at p = 0.25 average 5 correct. The single most likely count is near that mean, with probabilities tapering off on either side.