HMMT 二月 2007 · GEN2 赛 · 第 4 题

英文解析

4. [ 3 ] Three brothers Abel, Banach, and Gauss each have portable music players that can share musicwith each other. Initially, Abel has 9 songs, Banach has 6 songs, and Gauss has 3 songs, and none ofthese songs are the same. One day, Abel flips a coin to randomly choose one of his brothers and headds all of that brother’s songs to his collection. The next day, Banach flips a coin to randomly chooseone of his brothers and he adds all of that brother’s collection of songs to his collection. Finally, eachbrother randomly plays a song from his collection with each song in his collection being equally likelyto be chosen. What is the probability that they all play the same song?
   Answer: . If Abel copies Banach’s songs, this can never happen. Therefore, we consider only the 1
   cases where Abel copies Gauss’s songs. Since all brothers have Gauss’s set of songs, the probability that 288
   they play the same song is equivalent to the probability that they independently match whichever song
   Gauss chooses. Case 1: Abel copies Gauss and Banach copies Gauss (1/4 chance) – The probability ofsongs matching is then 1 / 12 · 1 / 9 . Case 2: Abel copies Gauss and Banach copies Abel (1/4 probability)
   – The probability of songs matching is then 1 / 12 · 1 / 18. We add the two probabilities together to get
   1 / 4 · 1 / 12 · (1 / 9 + 1 / 18) = 1 / 288 .