Corner Absorption From a 3x3 Center
On a $5 \times 5$ grid, label the columns $0$-$4$ and the rows $0$ (bottom) to $4$ (top). Pick a starting square uniformly at random from the central $3 \times 3$ block (columns
$-$3$, rows
$-$3$).
At each step the token moves diagonally upward: up-left or up-right, each with probability $\tfrac{1}{2}$. If one of those moves would take the token off the board, only the legal move is made (with probability
$). The walk ends when the token reaches row $4$.
What is the probability that the walk finishes in one of the two top corners -- column $0$ or column $4$ of row $4$? Show how you aggregate over all nine equally likely starting squares.
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