The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. It's one of the most widely used probability distributions in statistics, forming the foundation for hypothesis testing, quality control, and risk analysis. A binomial probability calculator computes the likelihood of observing exactly k successes in n trials.
The Formula
Where C(n,k) = n! / (k! × (n−k)!) is the number of ways to choose k successes from n trials. The complete binomial probability distribution shows the probability for every possible value of k from 0 to n.
Normal Approximation
When np ≥ 5 and n(1-p) ≥ 5, the binomial distribution can be approximated by a normal distribution with mean μ = np and standard deviation σ = √(np(1-p)). This normal approximation to the binomial is useful for large sample sizes and forms the basis for the z-test for proportions. Our calculator automatically overlays the normal curve when the conditions are satisfied.
Solving for n
Sometimes you need to find the minimum number of trials n needed to achieve a desired probability. For example: "How many times do I need to flip a coin to have a 95% chance of getting at least 5 heads?" Our solve for n calculator uses trial-and-error to find the smallest n such that P(X ≥ k) ≥ target probability.
Worked Example
You flip a fair coin 10 times. What's the probability of getting exactly 6 heads? Here n=10, k=6, p=0.5. C(10,6) = 10!/(6!×4!) = 210. So P(X=6) = 210 × 0.5⁶ × 0.5⁴ = 210 × 0.015625 × 0.0625 = 0.2051 ≈ 20.5%.
Distribution Statistics
The mean of a binomial distribution is E(X) = np, representing the expected number of successes. The variance is Var(X) = np(1-p), measuring the spread of the distribution. The standard deviation is the square root of the variance. For example, with n=100 and p=0.5, the expected number of heads is 50, with a standard deviation of 5 — meaning about 68% of experiments will yield between 45 and 55 heads.