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Unformatted text preview: ace i = 37 and roll k = 6. 37 + 6 = 43
but when we divide by 40 the remainder is 3, so p(37, 3) = r6 = 5/36.
This example is larger but has the same structure as the previous example.
Each row has the same entries but shift one unit to the right each time with the
number that goes o↵ the right edge emerging in the 0 column. This structure
implies that each entry in the row appears once in each column and hence the
sum of the entries in the column is 1, and the stationary distribution is uniform.
To check the hypothesis of the convergence theorem note that in four rolls you
can move forward by 8 to 48 squares, so p4 (i, j ) > 0 for all i and j .
Example 1.28. Real Monopoly has two complications:
• Square 30 is “Go to Jail,” which sends you to square 10. You can buy
your way out of jail but in the results we report below, we assume that
you are cheap. If you roll a double then you get out for free. If you don’t
get doubles in three tries you have to pay.
• There are three Chance squares at 7, 12, and 36...
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- Spring '10
- The Land