# Paper 2, Section II, 10E

(a) Alanya repeatedly rolls a fair six-sided die. What is the probability that the first number she rolls is a 1 , given that she rolls a 1 before she rolls a $6 ?$

(b) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a simple symmetric random walk on the integers starting at $x \in \mathbb{Z}$, that is,

$X_{n}=\left\{\begin{array}{cl} x & \text { if } n=0 \\ x+\sum_{i=1}^{n} Y_{i} & \text { if } n \geqslant 1 \end{array}\right.$

where $\left(Y_{n}\right)_{n \geqslant 1}$ is a sequence of IID random variables with $\mathbb{P}\left(Y_{n}=1\right)=\mathbb{P}\left(Y_{n}=-1\right)=\frac{1}{2}$. Let $T=\min \left\{n \geqslant 0: X_{n}=0\right\}$ be the time that the walk first hits 0 .

(i) Let $n$ be a positive integer. For $0, calculate the probability that the walk hits 0 before it hits $n$.

(ii) Let $x=1$ and let $A$ be the event that the walk hits 0 before it hits 3 . Find $\mathbb{P}\left(X_{1}=0 \mid A\right)$. Hence find $\mathbb{E}(T \mid A)$.

(iii) Let $x=1$ and let $B$ be the event that the walk hits 0 before it hits 4 . Find $\mathbb{E}(T \mid B)$.