The formula
P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ
What it means
The binomial counts successes across a fixed number of independent trials with the same probability each: heads in ten tosses, defective parts in a batch, correct answers on a multiple-choice test. The formula has two parts: the powers give the probability of one specific sequence with k successes, and the combination counts how many different ways those k successes can sit among the n trials.
How to work it out by hand
- Count how many trials there are in total and how likely each one is to succeed
- Work out C(n, k): how many ways the k successes can be arranged
- Multiply by p to the power k and by (1 − p) to the power n − k
- For the cumulative figures, add up the probabilities of all the cases you care about
What is worth knowing
The three conditions — fixed number of trials, independence and constant probability — are not a formality: draw cards without replacing them and the probability changes at every draw, which calls for the hypergeometric, not the binomial. When n is large and p small, the binomial closely resembles a Poisson with mean n·p, and when n is large with p in the middle, a normal; both approximations exist because the exact calculation was unworkable by hand, which is no longer the case here.