Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Discrete mathematics, continuous mathematics and probability / 7 / a

Let $(B_i)$ be a finite or countable partition of a sample space into events of positive probability. The events $A\cap B_i$ are disjoint and their union is $A$. Countable additivity and the definition of conditional probability therefore give the law of total probabilityCells of zero probability contribute zero through $\mathbb{P}(A\cap B_i)$ and can be omitted rather than assigning an undefined conditional probability.

The two definitions of conditional probability giveSubstitution proves Bayes' theorem:The positive probabilities assumed in the question make both conditional expressions defined.

Mutual independence means that for every nonempty $I\subseteq \{1,2,3\}$,Beyond the trivial single-event identities, this requires the three pairwise equalities and the triple equality $\mathbb{P}(A_1\cap A_2\cap A_3)=\mathbb{P}(A_1)\mathbb{P}(A_2)\mathbb{P}(A_3)$. Pairwise independence alone omits that last requirement.