In more detail
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k). The Combination C(n, k) counts which of the n trials are the successes, and the two powers give the chance of any one such pattern. It applies when four conditions hold: a fixed number of trials, two outcomes per trial, independent trials and the same success probability p each time. The mean is n × p and the standard deviation √(n × p × (1 − p)). Exactly 3 heads in 10 fair tosses has probability 120 / 1024 = 0.1172; at most 3 heads, 176 / 1024 = 0.1719.
Why it matters
It models pass/fail counts: defective parts in a batch, conversions in an A/B test, correct guesses on a multiple-choice quiz. It goes wrong when trials are not independent or p drifts, for example drawing without replacement from a small population. For "at least one" questions, compute "none" and subtract it from 1.
Example
A machine makes 5% defective parts and you inspect 20:
P(0 defective) = 0.95^20 = 0.3585
P(exactly 1) = 20 × 0.05 × 0.95^19 = 0.3774
P(at least 1) = 1 − 0.3585 = 0.6415
mean = 20 × 0.05 = 1 SD = √(20 × 0.05 × 0.95) = 0.975
Common confusion
The Poisson distribution counts events in an interval of time or space with no fixed number of trials. It approximates the binomial when n is large and p is small.