Consider the following game of Snakes and Ladders. The game is played by rolling

Question

Consider the following game of Snakes and Ladders. The game is played by rolling a 6-sided dice to move along the board. If you land at the bottom of a ladder, then you automatically climb to the higher number. If you land on a snake, then you must slide down to the lower number. Your last roll must land you exactly on the last number to win the game; if you roll too high a number, then you stay where you are. The position of a player at turn i, X_i, can be modeled as a Markov chain. Provide the probability transition matrix, P. Only include positions that a player can be on after any effect of snakes or ladders (hint, there are essentially only 22 positions)

Explanation / Answer

The answer is below

T 1 2 3 4 5 6 7 8 9 10 11 12 13 1 0.166667 0.166667 0.166667 0.166667 0.166667 0.166667 0 0 0 0 0 0 0 2 0 0 0.166667 0.166667 0.166667 0.166667 0.166667 0 0 0 0 0 0 3 0 0 0 0.166667 0.166667 0.166667 0.166667 0.166667 0 0 0 0 0 4 0 0 0 0 0.166667 0.166667 0.166667 0.166667 0.166667 0 0 0 0 5 0 0 0 0 0 0.166667 0.166667 0.166667 0.166667 0.166667 0 0 0 6 0 0 0 0 0 0 0.166667 0.166667 0.166667 0.166667 0.166667 0.166667 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 0 0 0

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