HMMT 二月 2001 · ADV 赛 · 第 6 题

英文解析

6. There are two red, two black, two white, and a positive but unknown number of bluesocks in a drawer. It is empirically determined that if two socks are taken from the drawerwithout replacement, the probability they are of the same color is . How many blue socks 1
   are there in the drawer?5
   Solution: Let the number of blue socks be x > 0. Then the probability of drawing ared sock from the drawer is and the probability of drawing a second red sock from the 2
   6+ x
   1 1
   drawer is = , so the probability of drawing two red socks from the drawer without
   6+ x − 1 5+ xreplacement is . This is the same as the probability of drawing two black socks 2
   (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 drawingx ( x − 1)
   two red, two black, two white, and two blue socks from the drawer: 3 + =2
   (6+ x )(5+ x ) (6+ x )(5+ x )
   x − x +6 12
   2 2 = . Cross-multiplying and distributing gives 5 x − 5 x + 30 = x + 11 x + 30, so
   (6+ x )(5+ x ) 5
   4 x − 16 x = 0, and x = 0 or 4. But since x > 0, there are 4 blue socks.2
   56 51 35 28