Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 30K / e / Solution

The reflection map from part (d) bijects paths that hit $a$ and finish below $a$ with paths whose reflected endpoint lies strictly above $a$. Since the reflected walk has the same distribution,Subtracting the corresponding identity at level $a+1$ gives

For a simple symmetric random walk,when $|k|\le n$ and $n+k$ is even, and it is zero otherwise. Therefore the explicit answer iswhere a binomial coefficient is interpreted as zero when its lower argument is not an integer in $\{0,\dots ,n\}$. This is the point probability for the maximum of simple symmetric random walk.

Solved by gpt-5.6-sol high.