Binomial Probability Calculator

Find the probability of k successes in n trials.

Number of trials (n)

Number of successes (k)

Success probability (p)

Between 0 and 1

Results update as you type.

Probability of exactly k

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Expected successes (n·p)—

How it works

The binomial distribution describes the probability of getting a certain number of successes in a fixed number of independent yes-or-no trials, and this calculator finds the exact probability of a specific outcome. You enter the number of trials, the number of successes you are interested in, and the probability of success on each trial, and it returns the probability of getting exactly that many successes, along with the expected number. Each trial must be independent and have the same success probability, like flipping a coin repeatedly or testing a batch of parts with a fixed defect rate. The formula combines how many ways the successes can be arranged among the trials with the probability of each specific arrangement. For example, the chance of exactly three heads in ten fair coin flips is about 11.7 percent. The binomial distribution is fundamental in probability and statistics, underlying quality control, genetics, clinical trials, polling, and any situation with repeated independent trials of a two-outcome event. The expected number of successes, simply the number of trials times the success probability, tells you the most likely long-run average. Enter the trials, successes, and probability to compute the binomial probability.

Formula

P(k) = C(n,k) · pᵏ · (1−p)^(n−k)

The binomial probability of exactly k successes in n independent trials multiplies three factors: the number of ways to choose which k of the n trials succeed, the probability of k successes each occurring, and the probability of the remaining trials all failing. The combination term counts the arrangements, while the powers of p and its complement give the probability of any one specific arrangement. The expected number of successes is n times p.

Source:Probability theory (binomial distribution)

We review all our tools from time to time to ensure accuracy.

Worked example

Number of trials (n):10Number of successes (k):3Success probability (p):0.5

P = C(10,3) × 0.5³ × 0.5⁷ = 120 × 0.5¹⁰ ≈ 0.1172 (about 11.7%).

How to use this calculator

1
Enter your values
Enter number of trials (n), number of successes (k) and success probability (p).
2
Adjust the options
Fine-tune any options to match your situation.
3
Read your result
Probability of exactly k updates instantly as you type — no button to press.

Frequently asked questions

What is a binomial probability?

A binomial probability is the chance of getting exactly a certain number of successes in a fixed number of independent trials, each with the same probability of success. It applies to situations with two outcomes per trial, like heads or tails. The binomial formula combines the number of possible arrangements with the probability of each to give the exact result.

What conditions must be met?

The binomial distribution requires a fixed number of trials, only two possible outcomes per trial, the same probability of success on every trial, and independence between trials so one outcome does not affect another. Coin flips, dice rolls checking for a specific face, and testing items with a constant defect rate all satisfy these conditions.

What does C(n,k) mean?

C(n,k), read as n choose k, is the number of different ways to select k successes among n trials, regardless of order. It is the binomial coefficient. For ten trials with three successes it equals 120, meaning there are 120 distinct arrangements of which three of the ten trials succeed. This count is a key part of the binomial formula.

What is the expected number of successes?

The expected number of successes is the number of trials multiplied by the probability of success, n times p. It is the long-run average you would expect over many repetitions. For ten flips of a fair coin, the expected number of heads is five. The actual result varies around this average according to the binomial distribution.

How is this different from finding at least k successes?

This calculator gives the probability of exactly k successes. To find the probability of at least k, or at most k, you would sum the binomial probabilities across the relevant range of outcomes. The single-outcome probability here is the building block; cumulative probabilities add several of these together to answer range-based questions.