Paper 2, Section II, 10E

Probability | Part IA, 2021

(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?6 ?

(b) Let (Xn)n0\left(X_{n}\right)_{n \geqslant 0} be a simple symmetric random walk on the integers starting at xZx \in \mathbb{Z}, that is,

Xn={x if n=0x+i=1nYi if n1X_{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 (Yn)n1\left(Y_{n}\right)_{n \geqslant 1} is a sequence of IID random variables with P(Yn=1)=P(Yn=1)=12\mathbb{P}\left(Y_{n}=1\right)=\mathbb{P}\left(Y_{n}=-1\right)=\frac{1}{2}. Let T=min{n0:Xn=0}T=\min \left\{n \geqslant 0: X_{n}=0\right\} be the time that the walk first hits 0 .

(i) Let nn be a positive integer. For 0<x<n0<x<n, calculate the probability that the walk hits 0 before it hits nn.

(ii) Let x=1x=1 and let AA be the event that the walk hits 0 before it hits 3 . Find P(X1=0A)\mathbb{P}\left(X_{1}=0 \mid A\right). Hence find E(TA)\mathbb{E}(T \mid A).

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

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