HMMT 十一月 2012 · THM 赛 · 第 8 题

英文解析

8. [ 4 ] In the game of rock-paper-scissors-lizard-Spock, rock defeats scissors and lizard, paper defeatsrock and Spock, scissors defeats paper and lizard, lizard defeats paper and Spock, and Spock defeatsrock and scissors, as shown in the below diagram. As before, if two players choose the same move,
   then there is a draw. If three people each play a game of rock-paper-scissors-lizard-Spock at the sametime by choosing one of the five moves at random, what is the probability that one player beats the other two?
   Scissors
   Spock Paper
   Lizard Rock
   Answer: Let the three players be A, B, C . Our answer will simply be the sum of the probability 12
   that A beats both B and C , the probability that B beats both C and A , and the probability that C25
   beats A and B , because these events are all mutually exclusive. By symmetry, these three probabilitiesare the same, so we only need to compute the probability that A beats both B and C . Given A ’splay, the probability that B ’s play loses to that of A is 2 / 5, and similarly for C . Thus, our answer is
   ( ) ( )
   2 2 12
   3 · · = .
   5 5 25