HMMT 二月 2009 · 冲刺赛 · 第 35 题

HMMT February 2009 — Guts Round — Problem 35

[ ≤ 25 ] Von Neumann’s Poker: The first step in Von Neumann’s game is selecting a random number on [0 , 1]. To generate this number, Chebby uses the factorial base: the number 0 .A A A A . . . stands 1 2 3 4 ∑ ∞ A n for , where each A is an integer between 0 and n , inclusive. n n =0 ( n +1)! Chebby has an infinite number of cards labeled 0 , 1 , 2 , . . . . He begins by putting cards 0 and 1 into a hat and drawing randomly to determine A . The card assigned A does not get reused. Chebby then 1 1 adds in card 2 and draws for A , and continues in this manner to determine the random number. At 2 each step, he only draws one card from two in the hat. Unfortunately, this method does not result in a uniform distribution. What is the expected value of Chebby’s final number? Your score on this problem will be the larger of 0 and b 25(1 − d ) c , where d is the positive difference between your answer and the correct answer.

解析

[ ≤ 25 ] Von Neumann’s Poker: The first step in Von Neumann’s game is selecting a random number on [0 , 1]. To generate this number, Chebby uses the factorial base: the number 0 .A A A A . . . stands 1 2 3 4 ∑ ∞ A n for , where each A is an integer between 0 and n , inclusive. n n =0 ( n +1)! Chebby has an infinite number of cards labeled 0 , 1 , 2 , . . . . He begins by putting cards 0 and 1 into a hat and drawing randomly to determine A . The card assigned A does not get reused. Chebby then 1 1 adds in card 2 and draws for A , and continues in this manner to determine the random number. At 2 each step, he only draws one card from two in the hat. Unfortunately, this method does not result in a uniform distribution. What is the expected value of Chebby’s final number? Your score on this problem will be the larger of 0 and b 25(1 − d ) c , where d is the positive difference between your answer and the correct answer. Answer: . 57196