HMMT 二月 2007 · COMB 赛 · 第 7 题

HMMT February 2007 — COMB Round — Problem 7

[ 5 ] Forty two cards are labeled with the natural numbers 1 through 42 and randomly shuffled into a stack. One by one, cards are taken off of the top of the stack until a card labeled with a prime number is removed. How many cards are removed on average?

解析

[ 5 ] Forty two cards are labeled with the natural numbers 1 through 42 and randomly shuffled into a stack. One by one, cards are taken off of the top of the stack until a card labeled with a prime number is removed. How many cards are removed on average? 43 Answer: . Note that there are 13 prime numbers amongst the cards. We may view these as 14 separating the remaining 29 cards into 14 groups of nonprimes - those appearing before the first prime, between the first and second, etc. Each of these groups is equally likely to appear first, so 29 / 14 nonprimes are removed on average. We are done since exactly one prime is always drawn.