HMMT 二月 2003 · COMB 赛 · 第 7 题
HMMT February 2003 — COMB Round — Problem 7
题目详情
英文原题
- You have infinitely many boxes, and you randomly put 3 balls into them. The boxesnare labeled 1 , 2 , . . . . Each ball has probability 1 / 2 of being put into box n . The ballsare placed independently of each other. What is the probability that some box willcontain at least 2 balls?
解析
英文解析
- You have infinitely many boxes, and you randomly put 3 balls into them. The boxesnare labeled 1 , 2 , . . . . Each ball has probability 1 / 2 of being put into box n . The ballsare placed independently of each other. What is the probability that some box willcontain at least 2 balls?
Solution: 5 / 7
Notice that the answer is the sum of the probabilities that boxes 1 , 2 , . . . , respectively,2
contain at least 2 balls, since those events are mutually exclusive. For box n , the probability of having at least 2 balls isn 2 n n 3 2 n 3 n n n
3[(1 / 2 ) (1 − 1 / 2 )] + (1 / 2 ) = 3 / 2 − 2 / 2 = 3 / 4 − 2 / 8 .
Summing to infinity using the geometric series formula, we get the answer (3 / 4) / (1 −
1 / 4) − (2 / 8) / (1 − 1 / 8), which is equal to 5 / 7.