Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 2 / 10E (Probability)

Ignore rolls showing $2,3,4,5$ and consider the first roll among $\{1,6\}$. By symmetry, the event $A$ that a 1 occurs before a 6 has probability $1/2$. The event that the very first roll is 1 has probability $1/6$ and is contained in $A$. Hence

Let $h(x)$ be the probability that the simple symmetric random walk starting at $x\in \{0,\dots ,n\}$ hits zero before $n$. The first-step recurrence isThe recurrence says that $h$ is affine, and the boundary values determine it:

Part (i) gives $\mathbb{P}(A)=2/3$. Since $\{X_1=0\}\subseteq A$ and has probability $1/2$,Conditioned on $A$, a walk at one moves to zero with probability $3/4$ and to two with probability $1/4$. A walk at two must next move to one, since a move to three would violate $A$. Iffirst-step analysis givesSolving,

For hitting zero before four, the success probabilities areConditioning each transition on eventual success givesThus the conditional expected hitting times satisfySolving this linear system gives