The king's random walk

Let the king wander, choosing a neighbour at random each step. Over many walks the fraction of time it ends on each square settles toward stable proportions. You estimate them empirically: run many simulated walks, tally the endings, and divide by the number of walks to get a proportion.

The empirical proportion approaches the true probability as the number of walks grows, an instance of the law of large numbers. Tabulating the tallies makes the pattern visible: central squares, with more neighbours feeding into them, tend to collect more visits than corners.