How it works
The binomial distribution gives the probability of a certain number of successes in a fixed number of independent trials, each with the same chance of success. Examples include the number of heads in 10 coin flips, or the number of defective parts in a batch of 50.
Pick exactly for a single count, at most for the cumulative probability up to k (like binomcdf), or at least for k or more successes.
The calculation uses logarithms of factorials so it stays accurate for large n. The chart shows the probability of each outcome.
Formula
P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ · Mean = np · SD = √(np(1 − p))
Questions people ask
When can I use the binomial distribution?
When there is a fixed number of trials, each trial has only two outcomes, the probability of success is the same every time, and trials are independent.
What is the chance of exactly 5 heads in 10 fair flips?
C(10,5) × 0.5¹⁰ = 252 ÷ 1,024 ≈ 24.61%. Even the most likely outcome happens less than a quarter of the time.
How does this relate to the normal distribution?
When np and n(1 − p) are both at least about 10, the binomial distribution is close to a normal distribution with mean np and standard deviation √(np(1 − p)).