Replacement and total probability

With replacement and random independent draws, category probabilities stay constant. Without replacement they change after each draw. For a partition B1,...,Bk, P(A)=∑P(A|Bj)P(Bj). Bayes rule reverses conditioning: P(Bj|A)=P(A|Bj)P(Bj)/P(A). The wet/dry season problem in T72 motivates this reversal. Write both stages explicitly to avoid reversing a conditional probability.