5.5.3—Binomial distribution

If X counts successes in n independent trials with constant success probability p, X~B(n,p) and P(X=r)=C(n,r)p^r(1−p)^{n−r}.

Check fixed n, two outcomes per trial, independence and constant p. Use complements or cumulative probabilities for “at least” and “at most” wording.

For X~B(10,0.3), P(X=2)=C(10,2)0.3²0.7⁸.

Binomial trials need not be equally likely across outcomes, but p must remain constant; sampling without replacement may break independence.