HMMT 二月 2003 · 冲刺赛 · 第 35 题

英文解析

35. A certain lottery has tickets labeled with the numbers 1 , 2 , 3 , . . . , 1000. The lottery isrun 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 ticketis randomly drawn (again without replacement), and so forth, until an odd numberis drawn. Then, every person whose ticket number was drawn (at any point in theprocess) 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, inrandom order, and awards a prize to the holder of the odd ticket drawn earliest and eacheven ticket drawn before it. Thus, the probability of your winning is the probabilitythat, 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 yourown 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.