2.II.9F

(a) Define the conditional probability $P(A \mid B)$ of the event $A$ given the event $B$. Let $\left\{B_{i}: 1 \leq i \leq n\right\}$ be a partition of the sample space $\Omega$ such that $P\left(B_{i}\right)>0$ for all $i$. Show that, if $P(A)>0$,

$P\left(B_{i} \mid A\right)=\frac{P\left(A \mid B_{i}\right) P\left(B_{i}\right)}{\sum_{j} P\left(A \mid B_{j}\right) P\left(B_{j}\right)} .$

(b) There are $n$ urns, the $r$ th of which contains $r-1$ red balls and $n-r$ blue balls. You pick an urn (uniformly) at random and remove two balls without replacement. Find the probability that the first ball is blue, and the conditional probability that the second ball is blue given that the first is blue. [You may assume that $\sum_{i=1}^{n-1} i(i-1)=\frac{1}{3} n(n-1)(n-2)$.]

(c) What is meant by saying that two events $A$ and $B$ are independent?

(d) Two fair dice are rolled. Let $A_{s}$ be the event that the sum of the numbers shown is $s$, and let $B_{i}$ be the event that the first die shows $i$. For what values of $s$ and $i$ are the two events $A_{s}, B_{i}$ independent?

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