HMMT 十一月 2017 · 团队赛 · 第 7 题
HMMT November 2017 — Team Round — Problem 7
题目详情
- [ 50 ] There are 12 students in a classroom; 6 of them are Democrats and 6 of them are Republicans. Every hour the students are randomly separated into four groups of three for political debates. If a group contains students from both parties, the minority in the group will change his/her political alignment to that of the majority at the end of the debate. What is the expected amount of time needed for all 12 students to have the same political alignment, in hours?
解析
- [ 50 ] There are 12 students in a classroom; 6 of them are Democrats and 6 of them are Republicans. Every hour the students are randomly separated into four groups of three for political debates. If a group contains students from both parties, the minority in the group will change his/her political alignment to that of the majority at the end of the debate. What is the expected amount of time needed for all 12 students to have the same political alignment, in hours? Proposed by: Yuan Yao 341 Answer: 54 When the party distribution is 6 − 6, the situation can change (to 3 − 9) only when a group of three contains three people from the same party, and the remaining three are distributed evenly across the other three groups (to be converted). To compute the probability, we assume that the groups and the members of the group are ordered (so there are 12! ways of grouping). There are 2 ways to choose the party, 4 ways to choose the group, 6 · 5 · 4 ways to choose the three members of the group, 9 · 6 · 3 ways to place the other three members of the party, and 6! ways to fill in the members of the other party. The probability is then 2 · 4 · 6 · 5 · 4 · 9 · 6 · 3 · 6! 2 · 4 · 6 · 5 · 4 · 9 · 6 · 3 18 = = . 12! 12 · 11 · 10 · 9 · 8 · 7 77 77 This means that the shift in distribution will happen in hours on average. 18 When the distribution is 3 − 9, the situation can change (to 0 − 12) only when the three members of the minority party are all in different groups. Using the similar method as above, there are 12 · 9 · 6 ways to place the three members and 9! ways to place the rest, so the probability is 12 · 9 · 6 · 9! 12 · 9 · 6 27 = = . 12! 12 · 11 · 10 55 55 This means that the shift in distribution will happen in hours on average. 27 77 55 By linearity of expectation, we can add up the two results and get that the expected value is + = 18 27 341 hours. 55