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HMMT 十一月 2016 · 冲刺赛 · 第 10 题

HMMT November 2016 — Guts Round — Problem 10

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 8 ] Michael is playing basketball. He makes 10% of his shots, and gets the ball back after 90% of hismissed shots. If he does not get the ball back he stops playing. What is the probability that Michaeleventually makes a shot?
解析

英文解析

  1. [ 8 ] Michael is playing basketball. He makes 10% of his shots, and gets the ball back after 90% of hismissed shots. If he does not get the ball back he stops playing. What is the probability that Michaeleventually makes a shot?
    Proposed by: Eshaan Nichani
    Answer: 10 / 19
    We find the probability Michael never makes a shot. We do casework on the number of shots Michael
    9 1 9
    takes. He takes only one shot with probability · (he misses with probability and does not get
    10 10 10
    1 9 9 9 1
    the ball back with probability ). Similarly, he takes two shots with probability · · · , three
    10 10 10 10 10
    ( ) ( ) 1 9
    ∑
    5 2 i − 1
    ·
    ∞
    9 1 9 1 9
    10 10
    shots with probability , and so on. So we want to sum · = = . Theni =181
    10 10 10 10 19
    1 −
    10100
    the probability Michael makes a shot is 1 − = .9
    1919