Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 5 / a

A partition of a set $\Omega$ is a collection of nonempty subsets whose union is $\Omega$ and which are pairwise disjoint:For use with probabilities, the $B_i$ must also be events in the sample space.

For a finite or countable partition of a set into events $B_i$, the law of total probability statesThe events $A\cap B_i$ are pairwise disjoint and have union $A$, so countable additivity proves the formula. Terms with $\mathbb{P}(B_i)=0$ contribute zero and are omitted to avoid undefined conditional probabilities.