HMMT 十一月 2020 · 冲刺赛 · 第 4 题

HMMT November 2020 — Guts Round — Problem 4

[6] Ainsley and Buddy play a game where they repeatedly roll a standard fair six-sided die. Ainsley wins if two multiples of 3 in a row are rolled before a non-multiple of 3 followed by a multiple of 3, and Buddy a wins otherwise. If the probability that Ainsley wins is for relatively prime positive integers a and b , b compute 100 a + b .

解析

[6] Ainsley and Buddy play a game where they repeatedly roll a standard fair six-sided die. Ainsley wins if two multiples of 3 in a row are rolled before a non-multiple of 3 followed by a multiple of 3, and a Buddy wins otherwise. If the probability that Ainsley wins is for relatively prime positive integers b a and b , compute 100 a + b . Proposed by: Dora Woodruff Answer: 109 Solution: We let X be the event of a multiple of 3 being rolled and Y be the event of a nonmultiple of 3 being rolled. In order for Ainsley to win, she needs event X to happen consecutively; meanwhile, Buddy just needs Y then X to occur. Thus, if Y occurs in the first two rolls, Buddy will be guaranteed to win, since the next time X happens, it will have been preceded by an X . Thus, the probability of A 2 winning is equivalent to the probability of X happening in each of the first two times, or (1 / 3) = 1 / 9.