Past exam of the year 1 of the computer science course of the University of Oxford / 2003 / Calculus, probability, linear algebra, discrete mathematics, logic and proof, and models of computation / 5 / a / ii / Solution

The partition theorem is the law of total probability. If $B_1,B_2,\dots$ form a finite or countable partition of a sample space into pairwise disjoint events with positive probabilities, then for any event $A$,Indeed, $A$ is the disjoint union of the events $A\cap B_j$, so countable additivity gives the formula after applying the definition of conditional probability. Zero-probability members can be omitted.