HMMT 二月 2003 · 冲刺赛 · 第 35 题

HMMT February 2003 — Guts Round — Problem 35

[12] A certain lottery has tickets labeled with the numbers 1 , 2 , 3 , . . . , 1000. The lottery is run as follows: First, a ticket is drawn at random. If the number on the ticket is odd, the drawing ends; if it is even, another ticket is randomly drawn (without replacement). If this new ticket has an odd number, the drawing ends; if it is even, another ticket is randomly drawn (again without replacement), and so forth, until an odd number is drawn. Then, every person whose ticket number was drawn (at any point in the process) wins a prize. You have ticket number 1000. What is the probability that you get a prize?

解析

A certain lottery has tickets labeled with the numbers 1 , 2 , 3 , . . . , 1000. The lottery is run as follows: First, a ticket is drawn at random. If the number on the ticket is odd, the drawing ends; if it is even, another ticket is randomly drawn (without replacement). If this new ticket has an odd number, the drawing ends; if it is even, another ticket is randomly drawn (again without replacement), and so forth, until an odd number is drawn. Then, every person whose ticket number was drawn (at any point in the process) wins a prize. You have ticket number 1000. What is the probability that you get a prize? Solution: 1 / 501 Notice that the outcome is the same as if the lottery instead draws all the tickets, in random order, and awards a prize to the holder of the odd ticket drawn earliest and each even ticket drawn before it. Thus, the probability of your winning is the probability that, in a random ordering of the tickets, your ticket precedes all the odd tickets. This, in turn, equals the probability that your ticket is the first in the setconsisting of your own and all odd-numbered tickets, irrespective of the other even-numbered tickets. Since all 501! orderings of these tickets are equally likely, the desired probability is 1 / 501.