HMMT 十一月 2016 · THM 赛 · 第 5 题

HMMT November 2016 — THM Round — Problem 5

Steph Curry is playing the following game and he wins if he has exactly 5 points at some time. Flip a fair coin. If heads, shoot a 3-point shot which is worth 3 points. If tails, shoot a free throw which is 1 worth 1 point. He makes of his 3-point shots and all of his free throws. Find the probability he will 2 win the game. (Note he keeps flipping the coin until he has exactly 5 or goes over 5 points) 6 3

解析

Steph Curry is playing the following game and he wins if he has exactly 5 points at some time. Flip a fair coin. If heads, shoot a 3-point shot which is worth 3 points. If tails, shoot a free throw which is 1 worth 1 point. He makes of his 3-point shots and all of his free throws. Find the probability he will 2 win the game. (Note he keeps flipping the coin until he has exactly 5 or goes over 5 points) Proposed by: Allen Liu 140 Answer: 243 If he misses the shot, then the state of the game is the same as before he flipped the coin. Since the 1 1 probability of making a free throw is and the probability of making a 3-point shot is . Therefore, 2 4 1 given that he earns some point, the probability it is a 3-point shot is . The possible ways of earning 3 140 32 4 4 4 points are 11111,113, 131, and 311, which have probabilities , , , and , which sum to . 243 27 27 27 243 6 3