Problem:
In a drawer Sandy has 5 pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the socks in the drawer. On Tuesday Sandy selects of the remaining socks at random and on Wednesday two of the remaining socks at random. The probability that Wednesday is the first day Sandy selects matching socks is , where and are relatively prime positive integers. Find .
Solution:
The probability of a match on Monday is , so the probability of a mismatch is . A mismatch on Tuesday can occur in one of three ways:
(a) On Tuesday Sandy selects both of the mismatching colors selected Monday. The probability of this is . In this case Sandy has three colors from which to select a pair on Wednesday.
(b) On Tuesday Sandy selects one of the colors selected Monday and a new color. The probability of this is . In this case Sandy has two colors from which to select a pair on Wednesday.
(c) On Tuesday Sandy selects two new colors. The probability of this is . In this case Sandy has only one color from which to select a pair on Wednesday.
The probability of achieving the first match on Wednesday is therefore
. The requested sum is .
The probability of Sandy getting a mismatch on Monday, a mismatch on Tuesday, and a match on Wednesday is the same as the probability of Sandy getting a match on Monday, a mismatch on Tuesday, and a mismatch on Wednesday. The latter probability is .
The problems on this page are the property of the MAA's American Mathematics Competitions