HMMT 二月 2008 · 冲刺赛 · 第 15 题
HMMT February 2008 — Guts Round — Problem 15
题目详情
英文原题
- [ 8 ] In a game show, Bob is faced with 7 doors, 2 of which hide prizes. After he chooses a door, thehost opens three other doors, of which one is hiding a prize. Bob chooses to switch to another door.
What is the probability that his new door is hiding a prize?
11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUNDth
解析
英文解析
- [ 8 ] In a game show, Bob is faced with 7 doors, 2 of which hide prizes. After he chooses a door, thehost opens three other doors, of which one is hiding a prize. Bob chooses to switch to another door.
What is the probability that his new door is hiding a prize?
Answer: If Bob initially chooses a door with a prize, then he will not find a prize by switching.53
With probability 5 / 7 his original door does not hide the prize. After the host opens the three doors,21
the remaining three doors have equal probability of hiding the prize. Therefore, the probability that
5 1 5
Bob finds the prize is × = .
7 3 21
Remark: This problem can be easily recognized as a variation of the classic Monty Hall problem.
11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUNDth