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HMMT 二月 2010 · 冲刺赛 · 第 9 题

HMMT February 2010 — Guts Round — Problem 9

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 6 ] Indecisive Andy starts out at the midpoint of the 1-unit-long segment HT . He flips 2010 coins. Oneach flip, if the coin is heads, he moves halfway towards endpoint H , and if the coin is tails, he moveshalfway towards endpoint T . After his 2010 moves, what is the expected distance between Andy andthe midpoint of HT ?
    13 ANNUAL HARVARD-MIT MATHEMATICS TOURNAMENT, 20 FEBRUARY 2010 — GUTS ROUNDth
解析

英文解析

  1. [ 6 ] Indecisive Andy starts out at the midpoint of the 1-unit-long segment HT . He flips 2010 coins. Oneach flip, if the coin is heads, he moves halfway towards endpoint H , and if the coin is tails, he moveshalfway towards endpoint T . After his 2010 moves, what is the expected distance between Andy andthe midpoint of HT ?
    Answer: Let Andy’s position be x units from the H end after 2009 flips. If Any moves towards 1
    x 1 − x 4
    the H end, he ends up at , a distance of from the midpoint. If Andy moves towards the T end,
    2 2
    1+ x xhe ends up at , a distance of from the midpoint. His expected distance from the midpoint is then
    2 2
    1 − x x
    1+
    2 2 = .
    2 4
    Since this does not depend on x , is the answer.1
    13 ANNUAL HARVARD-MIT MATHEMATICS TOURNAMENT, 20 FEBRUARY 2010 — GUTS ROUNDth