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

$$\mathbb{P}\left(B_{j} \mid A\right)=\frac{\mathbb{P}\left(A \mid B_{j}\right) \mathbb{P}\left(B_{j}\right)}{\sum_{k=1}^{n} \mathbb{P}\left(A \mid B_{k}\right) \mathbb{P}\left(B_{k}\right)}$$

(ii) There are $n$ urns, the $r$ th of which contains $r-1$ red balls and $n-r$ blue balls. Alice picks an urn (uniformly) at random and removes 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, if you wish, that $\sum_{i=1}^{n-1} i(i-1)=\frac{1}{3} n(n-1)(n-2)$.]

(b) (i) What is meant by saying that two events $A$ and $B$ are independent? Two fair (6-sided) dice are rolled. Let $A_{t}$ be the event that the sum of the numbers shown is $t$, and let $B_{i}$ be the event that the first die shows $i$. For what values of $t$ and $i$ are the two events $A_{t}$ and $B_{i}$ independent?

(ii) The casino at Monte Corona features the following game: three coins each show heads with probability $3 / 5$ and tails otherwise. The first counts 10 points for a head and 2 for a tail; the second counts 4 points for both a head and a tail; and the third counts 3 points for a head and 20 for a tail. You and your opponent each choose a coin. You cannot both choose the same coin. Each of you tosses your coin once and the person with the larger score wins the jackpot. Would you prefer to be the first or the second to choose a coin?