HMMT 十一月 2013 · 团队赛 · 第 2 题

英文解析

2. [ 4 ] Gary plays the following game with a fair n -sided die whose faces are labeled with the positiveintegers between 1 and n , inclusive: if n = 1, he stops; otherwise he rolls the die, and starts over with ak -sided die, where k is the number his n -sided die lands on. (In particular, if he gets k = 1, he will stoprolling the die.) If he starts out with a 6-sided die, what is the expected number of rolls he makes?
   Answer: If we let a be the expected number of rolls starting with an n -sided die, we 197
   60 n
   ∑
   1 nsee immediately that a = 0, and a = 1 + a for n > 1. Thus a = 2, and for n ≥ 3,
   1 n i 2
   i =1
   ∑nn − 1
   1 n − 1 1 1
   a = 1 + a + ( a − 1), or a = a + . Thus a = 1 + for n ≥ 2, so a =
   n n n − 1 n n − 1 n 6
   i =1
   n n n − 1 i
   60+30+20+15+12 197
   1 + = .
   60 60