HMMT 十一月 2008 · 冲刺赛 · 第 22 题

HMMT November 2008 — Guts Round — Problem 22

[ 12 ] Sandra the Maverick has 5 pairs of shoes in a drawer, each pair a different color. Every day for 5 days, Sandra takes two shoes out and throws them out the window. If they are the same color, she treats herself to a practice problem from a past HMMT. What is the expected value (average number) of practice problems she gets to do? 2 2 2 2 ( x − 4) y y x

解析

[ 12 ] Sandra the Maverick has 5 pairs of shoes in a drawer, each pair a different color. Every day for 5 days, Sandra takes two shoes out and throws them out the window. If they are the same color, she treats herself to a practice problem from a past HMMT. What is the expected value (average number) of practice problems she gets to do? 5 1 Answer: On any given day, there is a chance that the second shoe that Sandra chooses makes 9 9 1 a pair with the first shoe she chose. Thus the average number of problems she does in a day is , so, 9 5 by the linearity of expectation, she does problems total, on average. 9 2 2 2 2 ( x − 4) y y x