HMMT 十一月 2010 · 冲刺赛 · 第 21 题

英文解析

21. [ 11 ] George, Jeff, Brian, and Travis decide to play a game of hot potato. They begin by arrangingthemselves clockwise in a circle in that order. George and Jeff both start with a hot potato. On histurn, a player gives a hot potato (if he has one) to a randomly chosen player among the other three (ifa player has two hot potatoes on his turn, he only passes one). If George goes first, and play proceedesclockwise, what is the probability that Travis has a hot potato after each player takes one turn?
    Answer: Notice that Travis can only have the hot potato at the end if he has two potatoes 5
    before his turn. A little bit of casework shows that this can only happen when 27
    Case 1 : George gives Travis his potato, while Jeff gives Brian his potato, which in then goes to Travis.
    ( )
    1 13
    The probability of this occuring is =
    3 27
    Case 2 : George gives Travis his potato, while Jeff gives Travis his potato. The probability of this
    ( )
    1 12
    occuring is =
    3 9
    Case 3 : George gives Brian his potato, Jeff gives Travis his potato, and then Brian gives Travis hispotato. The probability of this occuring is 1
    Because these events are all disjoint, the probability that Travis ends up with the hot potato is 527
    2127