Formula for Conditional Probability

The two worked examples in the previous section suggest a general shortcut: instead of listing the reduced sample space every time, conditional probability can be written directly in terms of ordinary probabilities. For a finite sample space $S$ with equally likely outcomes, once event $B$ is known to have occurred, the effective sample space shrinks to the $n(B)$ outcomes of $B$ (the reduced sample space). Among those, only the outcomes that also belong to $A$ - namely the $n(A\cap B)$ outcomes in $A\cap B$ - are favourable for $A$. So by the classical definition applied to this reduced space, $P(A/B)=\frac{n(A\cap B)}{n(B)}=\frac{n(A\cap B)/n(S)}{n(B)/n(S)}=\frac{P(A\cap B)}{P(B)},\qquad P(B)\ne 0.$ By the same reasoning with the roles of $A$ and $B$ swapped, $P(B/A)=\frac{P(A\cap B)}{P(A)},\qquad P(A)\ne 0.$ …