Question
Prove that if $H_{i}$ are pairwise disjoint events such that $\bigcup_{i=1}^{\infty} H_{i}=\Omega$ $P\left(H_{i}\right) \neq 0$, then$$P(A)=\sum_{i=1}^{\infty} P\left(A \mid H_{i}\right) P\left(H_{i}\right)$$
Step 1
Since the events $H_i$ are pairwise disjoint and their union is the entire sample space $\Omega$, they form a partition of $\Omega$. In other words, every outcome in $\Omega$ belongs to exactly one of the events $H_i$. Show more…
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