13. Let us consider the infinite integer lattice and assume that having reached a point $(x, n)$ or $(n, y)$, the particle continues moving east and north following the rules of the game. The required probability $p_{k}$ is equal to the probability of getting to one of the points $E_{1}(n, n+k), E_{2}(n+k, n)$, but without passing through $(n, n+k-1)$ or $(n+k-1, n)$. Thus $p$ is equal to the probability $p_{1}$ of getting to $E_{1}(n, n+k)$ via $D_{1}(n-1, n+k)$ plus the probability $p_{2}$ of getting to $E_{2}(n+k, n)$ via $D_{2}(n+k, n-1)$. Both $p_{1}$ and $p_{2}$ are easily seen to be equal to $\binom{2 n+k-1}{n-1} 2^{-2 n-k}$, and therefore $p=\binom{2 n+k-1}{n-1} 2^{-2 n-k+1}$.
13. (FRG 3) A particle moves from $(0,0)$ to $(n, n)$ directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At $(n, y), y<n$, it stays there if a head comes up and at $(x, n)$, $x<n$, it stays there if a tail comes up. Let $k$ be a fixed positive integer. Find the probability that the particle needs exactly $2 n+k$ tosses to reach $(n, n)$.