Lessons from Knight Probability

$1.$ State includes position and moves remaining. Both affect the probability.

$2.$ Transition sums weighted contributions. Each of 8 moves contributes $1/8$ of its probability.

$3.$ Off-board states return $0$. They're absorbing states where the process "fails." This pattern repeats in almost all probability DP: state captures position and progress, transitions are weighted sums, and boundaries return $0$ or $1$.