(a) State the axioms that must be satisfied by a probability measure $\mathbb{P}$ on a probability space $\Omega$.

Let $A$ and $B$ be events with $\mathbb{P}(B)>0$. Define the conditional probability $\mathbb{P}(A \mid B)$.

Let $B_{1}, B_{2}, \ldots$ be pairwise disjoint events with $\mathbb{P}\left(B_{i}\right)>0$ for all $i$ and $\Omega=\cup_{i=1}^{\infty} B_{i}$. Starting from the axioms, show that

$$\mathbb{P}(A)=\sum_{i=1}^{\infty} \mathbb{P}\left(A \mid B_{i}\right) \mathbb{P}\left(B_{i}\right)$$

and deduce Bayes' theorem.

(b) Two identical urns contain white balls and black balls. Urn I contains 45 white balls and 30 black balls. Urn II contains 12 white balls and 36 black balls. You do not know which urn is which.

(i) Suppose you select an urn and draw one ball at random from it. The ball is white. What is the probability that you selected Urn I?

(ii) Suppose instead you draw one ball at random from each urn. One of the balls is white and one is black. What is the probability that the white ball came from Urn I?

(c) Now suppose there are $n$ identical urns containing white balls and black balls, and again you do not know which urn is which. Each urn contains 1 white ball. The $i$ th urn contains $2^{i}-1$ black balls $(1 \leqslant i \leqslant n)$. You select an urn and draw one ball at random from it. The ball is white. Let $p(n)$ be the probability that if you replace this ball and again draw a ball at random from the same urn then the ball drawn on the second occasion is also white. Show that $p(n) \rightarrow \frac{1}{3}$ as $n \rightarrow \infty$