HMMT 十一月 2016 · 冲刺赛 · 第 10 题

HMMT November 2016 — Guts Round — Problem 10

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

解析

[ 8 ] Michael is playing basketball. He makes 10% of his shots, and gets the ball back after 90% of his missed shots. If he does not get the ball back he stops playing. What is the probability that Michael eventually 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 · = = . Then 81 i =1 10 10 10 10 19 1 − 100 10 9 the probability Michael makes a shot is 1 − = . 19 19