HMMT 二月 2001 · ADV 赛 · 第 6 题

HMMT February 2001 — ADV Round — Problem 6

There are two red, two black, two white, and a positive but unknown number of blue socks in a drawer. It is empirically determined that if two socks are taken from the drawer 1 without replacement, the probability they are of the same color is . How many blue socks 5 are there in the drawer? 56 51 35 28

解析

There are two red, two black, two white, and a positive but unknown number of blue socks in a drawer. It is empirically determined that if two socks are taken from the drawer 1 without replacement, the probability they are of the same color is . How many blue socks 5 are there in the drawer? Solution: Let the number of blue socks be x > 0. Then the probability of drawing a 2 red sock from the drawer is and the probability of drawing a second red sock from the 6+ x 1 1 drawer is = , so the probability of drawing two red socks from the drawer without 6+ x − 1 5+ x 2 replacement is . This is the same as the probability of drawing two black socks (6+ x )(5+ x ) from the drawer and the same as the probability of drawing two white socks from the drawer. x ( x − 1) The probability of drawing two blue socks from the drawer, similarly, is . Thus the (6+ x )(5+ x ) probability of drawing two socks of the same color is the sum of the probabilities of drawing x ( x − 1) 2 two red, two black, two white, and two blue socks from the drawer: 3 + = (6+ x )(5+ x ) (6+ x )(5+ x ) 2 x − x +6 1 2 2 = . Cross-multiplying and distributing gives 5 x − 5 x + 30 = x + 11 x + 30, so (6+ x )(5+ x ) 5 2 4 x − 16 x = 0, and x = 0 or 4. But since x > 0, there are 4 blue socks. 56 51 35 28