Alice, Betty and Cathy have a die with 12 faces numbered from 1 to 12; every number has an equal outcome probability of

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Alice Betty And Cathy Have A Die With 12 Faces Numbered From 1 To 12 Every Number Has An Equal Outcome Probability Of 1 (140.15 KiB) Viewed 32 times

Alice, Betty and Cathy have a die with 12 faces numbered from 1 to 12; every number has an equal outcome probability of outcome. They toss the die in turn until one of them wins by getting the first 1. Alice's probability of winning in the first round is P(1) = 1) 1 12 Her probability of winning in the second round is the probability that she tosses 1 after everyone tossed something else in the first round, i.e. 11 11 1 P(*, *, *,1) 113 11 12 12 12 12 124 Her probability of winning in the third round is the probability that she tosses 1 after everyone tossed something else in the first and second rounds, i.e. 116 P(*, *,*, 1) 127 And so on. Assuming that the players are tireless, Alice's probability of winning the game is 1 113 116 + + 12 124 127 +... This series converges to the value Betty's probability of winning the game is Cathy's probability of winning the game is