Calculate the probability of exactly k successes in n independent trials (binomial probability).
This calculates binomial probability — the chance of getting exactly k successes across n independent trials, each with the same probability p of success. Formula: P = C(n,k) × p^k × (1−p)^(n−k), where C(n,k) is the number of ways to arrange k successes among n trials.
'Independent' is the key requirement — each trial's outcome can't influence the next, which holds for coin flips and dice rolls but not for situations like drawing cards without replacement.
This is the standard tool for questions like 'what's the chance of exactly 3 heads in 5 coin flips' or 'what's the chance of exactly 2 defective parts in a batch of 20 with a known defect rate.'
The chance of exactly 3 heads in 5 fair coin flips (p = 0.5): C(5,3) × 0.5³ × 0.5² = 10 × 0.125 × 0.25 = 31.25%.
Calculate this for each value from k up to n and add the results together, or subtract the probability of fewer than k successes from 1 — this tool gives the exact-k case as the building block.
Yes — for example, the probability of rolling exactly two 6s in four rolls of a die uses p = 1/6 (≈0.1667) as the per-trial success probability.
Drawing cards without replacement changes the probability on each draw, violating the independence assumption this formula requires — that situation needs hypergeometric probability instead.