Chapter 11.3, Problem 50E

Proof of Formulas Given events A and B within the sample space S. the following sequence of steps establishes formulas that can be used to compute conditional probabilities. Justify each statement.

(a) $P(AandB)=P(A)\cdot P(B|A)$

(b) Therefore, $P(B|A)=\frac{P(AandB)}{P(A)}.$

(c) Therefore. $P(B|A)=\frac{n(A\text{and}B)/n(S)}{n(A)/n(S)}.$

(d) Therefore. $P(B|A)=\frac{n(A\text{and}B)}{n(A)}$