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Stochastic matrix - Wikipedia https://en.wikipedia.org/wiki/Stochastic_matrix
Suppose you have a timer and a row of five adjacent boxes, with a cat in the first box and a mouse in the fifth box at time zero. The cat and the mouse both jump to a random adjacent box when the timer advances. E.g. if the cat is in the second box and the mouse in the fourth one, the probability is one fourth that the cat will be in the first box and the mouse in the fifth after the timer advances. If the cat is in the first box and the mouse in the fifth one, the probability is one that the cat will be in box two and the mouse will be in box four after the timer advances. The cat eats the mouse if both end up in the same box, at which time the game ends. The random variable K gives the number of time steps the mouse stays in the game.
The Markov chain that represents this game contains the following five states specified by the combination of positions (cat,mouse). Note that while a naive enumeration of states would list 25 states, many are impossible either because the mouse can never have a lower index than the cat (as that would mean the mouse occupied the cat's box and survived to move past it), or because the sum of the two indices will always have even parity. In addition, the 3 possible states that lead to the mouse's death are combined into one:
- State 1: (1,3)
- State 2: (1,5)
- State 3: (2,4)
- State 4: (3,5)
- State 5: game over: (2,2), (3,3) & (4,4).