HMMT 十一月 2010 · 冲刺赛 · 第 1 题

英文解析

1. [ 5 ] David, Delong, and Justin each showed up to a problem writing session at a random time duringthe session. If David arrived before Delong, what is the probability that he also arrived before Justin?
   Answer: Let t be the time that David arrives, let t be the time that Delong arrives, and let t be 2
   1 2 3
   the time that Justin arrives. We can assume that all times are pairwise distinct because the probability 3
   of any two being equal is zero. Because the times were originally random and independent before wewere given any information, then all orders t < t < t , t < t < t , t < t < t , t < t < t ,
   1 2 3 1 3 2 3 1 2 2 1 3
   t < t < t , t < t < t must all be equally likely. Since we are given that t < t , then we only have
   2 3 1 3 2 1 1 2
   the first three cases to consider, and t < t in two cases of these three. Thus, the desired probability
   1 3
   is .2 3