🎯 Binomial Probability Calculator
Compute binomial probabilities for n trials with success probability p, exactly P(X=k), P(X≤k), P(X≥k), plus the mean and standard deviation.
P(X = 12)
0.12013435
12.0134%
P(X ≤ 12)
0.86841202
86.8412%
P(X ≥ 12)
0.25172234
25.1722%
P(X < 12)
0.74827766
74.8278%
P(X > 12)
0.13158798
13.1588%
10
Mean (np)
2.2361
Std. dev. √(np(1−p))
125970
C(20,12) exact
Binomial coefficients are computed with exact BigInt arithmetic, no floating-point overflow at large n. 🔒 Computed in your browser.
How the binomial probability calculator works
For n independent trials each with success probability p, the binomial distribution gives the chance of exactly k successes: P(X=k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ. The three parts multiply the number of ways to arrange k successes among n trials, C(n,k), by the probability of those k successes, pᵏ, and of the remaining n−k failures, (1−p)ⁿ⁻ᵏ. This tool computes the binomial coefficient C(n,k) with exact BigInt arithmetic, so results stay precise even for large n, and reports the exact, at-most, and at-least probabilities along with the mean (np) and standard deviation (√(np(1−p))). Worked example: the chance of exactly 3 heads in 5 fair coin flips is C(5,3)·0.5³·0.5² = 10 × 0.03125 = 0.3125.
On brand with our exact-math tools: the binomial coefficients use BigInt, so there is no floating-point overflow or rounding that plagues naive implementations at large n.
Frequently asked questions
What is the binomial probability formula?
P(X=k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ, where n is the number of trials, p the probability of success on each, k the number of successes, and C(n,k) the binomial coefficient (n choose k).
What is the difference between P(X=k), P(X≤k) and P(X≥k)?
P(X=k) is exactly k successes; P(X≤k) is k or fewer (the cumulative probability); P(X≥k) is k or more (1 − P(X≤k−1)). This tool reports all three.
When can I use the binomial distribution?
When you have a fixed number of independent trials (n), each with two outcomes and the same success probability (p), for example coin flips, defect rates, or yes/no survey responses.
What are the mean and standard deviation?
The mean (expected number of successes) is np, and the standard deviation is √(np(1−p)). For n=20, p=0.5 that is a mean of 10 and SD of about 2.24.
Why use BigInt coefficients?
Binomial coefficients grow astronomically, C(100,50) has 30 digits. Computing them with ordinary floating-point numbers loses precision; BigInt keeps the calculation exact.
Is this the same as a normal approximation?
No. This computes the exact binomial probability. For very large n the normal approximation is often used, but here you get the precise value.
What is the binomial coefficient C(n,k)?
It counts the number of ways to choose k items from n without regard to order: C(n,k) = n! ÷ (k!·(n−k)!). For n=5, k=3 that is 120 ÷ (6 × 2) = 10, the ten distinct positions the 3 successes can occupy among 5 trials.
What conditions must hold for a binomial distribution?
Four: a fixed number of trials n, only two outcomes per trial (success/failure), the same success probability p on every trial, and independence between trials. If p changes or trials influence each other, the binomial model does not apply.