Proof: Total Probability
Given that the collection \(A_i\) is a parition of \(\Omega\), for any set \(B\subseteq \Omega\),
\[B = \bigcup_{i}^{} A_i \cap B\]Also thanks to \(A_i\)’s partitioning \(\Omega\), the sets that constitue \(B\) as above are mutually exclusive. Given this, by the additivity axiom of probability, we have:
\[P(B) = \sum_{i}^{} P(A_i \cap B)\]The rest follows from the definition of conditional probability .