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MIT6_042JS10_lec39_sol

# MIT6_042JS10_lec39_sol - Massachusetts Institute of...

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Unformatted text preview: Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science May 12 Prof. Albert R. Meyer revised May 10, 2010, 678 minutes Solutions to In-Class Problems Week 14, Wed. Problem 1. A gambler is placing \$1 bets on the “1st dozen” in roulette. This bet wins when a number from one to twelve comes in, and then the gambler gets his \$1 back plus \$3 more. Recall that there are 38 numbers on the roulette wheel. The gambler’s initial stake in \$ n and his target is \$ T . He will keep betting until he runs out of money (“goes broke”) or reachs his target. Let w n be the probability of the gambler winning, that is, reaching target \$ T before going broke. (a) Write a linear recurrence for w n ; you need not solve the recurrence. Solution. The probability of winning a bet is 12 / 38 . Thus, by the Law of Total Probability ?? , w n = Pr { win starting with \$ n | won first bet } · Pr { won first bet } + Pr { win starting with \$ n | lost first bet } · Pr = Pr { win starting with \$ n + 3 } · Pr { won first bet } + Pr { win starting with \$...
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MIT6_042JS10_lec39_sol - Massachusetts Institute of...

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