Binomial Probabilities

A binomial setting has a fixed number of trials n, two outcomes per trial (success or failure), a constant success probability p, and independence between trials. For any one specific sequence with exactly k successes and n-k failures, the probability is p^k times (1-p)^n-k, since independent probabilities multiply. But that sequence is just one way to land exactly k successes among n trials — the successes could fall on any k of the n positions. The count of those arrangements is n choose k, written C(n,k) or nCk, equal to n! / (k!(n-k)!). Multiplying the per-sequence probability by the number of sequences gives the full binomial formula: P(X=k) = C(n,k) p^k (1-p)^n-k.

n choose k comes from counting combinations: choose which k of the n trial slots are successes, ignoring order, since swapping two successes doesn't create a new outcome. That's why it sits out front — it corrects for the fact that many different orderings all produce the same count of successes, and each ordering has the same probability p^k(1-p)^n-k.

This formula is the engine behind the binomial distribution, which you'll soon use to find means (np) and standard deviations (√(np(1-p))), and which later gets approximated by a normal curve when n is large. Cumulative questions — at most, at least, more than — just mean summing P(X=k) over the right set of k values, since outcomes for different k are mutually exclusive.