七个小矮人睡床
seven dwarfs
题目详情
Each of the seven dwarfs has his own bed in a common dormitory. Every night, they retire to bed one at a time, always in the same sequential order according to their age. On a particular evening, the youngest dwarf, who always retires first, has had too much to drink. He randomly chooses one of the seven beds to fall asleep in. As each of the other dwarfs retires, he chooses his own bed if it is not occupied, and otherwise
randomly chooses another unoccupied bed. For each of the other dwarfs, what is the probability that he will sleep in his own bed?
解析
这是经典“飞机随机入座”模型。
对第 个进入的矮人（按顺序从 1 到 7）：
- 第 1 个（最年轻）随机选床，睡自己床的概率为 。
- 对 ，第 个睡自己床的概率为
具体数值：
- 第 2 个：
- 第 3 个：
- 第 4 个：
- 第 5 个：
- 第 6 个：
- 第 7 个：。