Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability (resit) / 7 / a

For events $A,B$ with $\mathbb{P}(B)>0$, the conditional probability isThis renormalizes the probability of $A$ within $B$. The elementary ratio definition does not assign a value when $\mathbb{P}(B)=0$.

Since the $B_i$ form a partition of a set of the sample space, the events $A\cap B_i$ are pairwise disjoint and their set union is $A$. By countable additivity of probability, followed by the definition of conditional probability,Every ratio is legitimate because each $\mathbb{P}(B_i)>0$. This proves the countable law of total probability.