HMMT 二月 2009 · 冲刺赛 · 第 35 题
HMMT February 2009 — Guts Round — Problem 35
题目详情
英文原题
- [ ≤ 25 ] Von Neumann’s Poker: The first step in Von Neumann’s game is selecting a random numberon [0 , 1]. To generate this number, Chebby uses the factorial base: the number 0 .A A A A . . . stands
1 2 3 4
∑
∞
for , where each A is an integer between 0 and n , inclusive.n ann =0
( n +1)!
Chebby has an infinite number of cards labeled 0 , 1 , 2 , . . . . He begins by putting cards 0 and 1 into ahat 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. Ateach step, he only draws one card from two in the hat.2
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 differencebetween your answer and the correct answer.
解析
英文解析
- [ ≤ 25 ] Von Neumann’s Poker: The first step in Von Neumann’s game is selecting a random numberon [0 , 1]. To generate this number, Chebby uses the factorial base: the number 0 .A A A A . . . stands
1 2 3 4
∑
∞
for , where each A is an integer between 0 and n , inclusive.n ann =0
( n +1)!
Chebby has an infinite number of cards labeled 0 , 1 , 2 , . . . . He begins by putting cards 0 and 1 into ahat 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. Ateach step, he only draws one card from two in the hat.2
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 differencebetween your answer and the correct answer.
Answer: . 57196