HMMT 十一月 2021 · GEN 赛 · 第 6 题

HMMT November 2021 — GEN Round — Problem 6

Mario has a deck of seven pairs of matching number cards and two pairs of matching Jokers, for a total of 18 cards. He shuffles the deck, then draws the cards from the top one by one until he holds a pair of matching Jokers. The expected number of complete pairs that Mario holds at the end (including the m Jokers) is , where m, n are positive integers and gcd( m, n ) = 1. Find 100 m + n . n

解析

Mario has a deck of seven pairs of matching number cards and two pairs of matching Jokers, for a total of 18 cards. He shuffles the deck, then draws the cards from the top one by one until he holds a pair of matching Jokers. The expected number of complete pairs that Mario holds at the end (including the m Jokers) is , where m, n are positive integers and gcd( m, n ) = 1. Find 100 m + n . n Proposed by: Sean Li Answer: 1003 Solution: Considering ordering the nine pairs by the time they are first complete. Since the pairs are treated equally by the drawing process, this ordering is a uniform ordering. Therefore the problem becomes the following: consider ordering 7 N’s and 2 J’s randomly. What is the expected position of the first J? We may solve this by linearity of expectation. Every N has exactly a 1 / 3 chance of being in front of the 2 J’s, so the expected number of N’s before the first J is 7 / 3. Thus the expected position of the first J is 7 / 3 + 1 = 10 / 3.