Question

The game of monopoly is played on a square board with 10 spaces per side (so 40 in total). A player starts at the space marked "go" and on each turn rolls two sided dice, advancing forward a number of spaces equal to the sum of the two numbers that lands face up. Let P be the probability that a player after they "loop around" the board 10 times, lands on the space "Boardwalk" (the 40th square) before they "loop around" the board an eleventh time. Which integer is closest to the value 20P?

Answer 1

make a spreadsheet having a row for each square on each turn of the board.

We can easily see that square 5 is the Reading Railroad on the first turn and 45 on the second turn and so on.

place the probability we hit the cell , in each cell.

Then , the chance in cell n is 1/36 .

the probability of n−2 plus 2/36 the probability of cell n−3 and so on.

If we start with twelve blank rows at the start we can use the same formula because the probability we hit cell −1 is zero. The chance we hit cell 0, the original GO, is 1.

Alternative informal approach is given below

formalize in the language of Markov chains, and note the average roll is 7, so we hit 1/7 of the squares. The original blip of 1 for the starting square washes out immediately.

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