HMMT 十一月 2016 · THM 赛 · 第 5 题
HMMT November 2016 — THM Round — Problem 5
题目详情
英文原题
- Steph Curry is playing the following game and he wins if he has exactly 5 points at some time. Flip afair coin. If heads, shoot a 3-point shot which is worth 3 points. If tails, shoot a free throw which isworth 1 point. He makes of his 3-point shots and all of his free throws. Find the probability he will 1
win the game. (Note he keeps flipping the coin until he has exactly 5 or goes over 5 points)2
6 3
解析
英文解析
- Steph Curry is playing the following game and he wins if he has exactly 5 points at some time. Flip afair coin. If heads, shoot a 3-point shot which is worth 3 points. If tails, shoot a free throw which isworth 1 point. He makes of his 3-point shots and all of his free throws. Find the probability he will 1
win the game. (Note he keeps flipping the coin until he has exactly 5 or goes over 5 points)2
Proposed by: Allen Liu
Answer:140
If he misses the shot, then the state of the game is the same as before he flipped the coin. Since the 243
1 1
probability of making a free throw is and the probability of making a 3-point shot is . Therefore,
2 4
given that he earns some point, the probability it is a 3-point shot is . The possible ways of earning 1
1403
32 4 4 4
points are 11111,113, 131, and 311, which have probabilities , , , and , which sum to .
243 27 27 27
6 3243