三只青蛙同点相遇的期望时间
Three frogs
题目详情
Three frogs are jumping on the vertices of an equilateral triangle. A vertex can be occupied by more than one frog. Every minute, each frog jumps from the vertex where it is located to one of the other two vertices, each being equally likely. The frogs choose where to jump independently of each other. If initially each vertex contains exactly one frog, how long does it take on average for all of the frogs to meet at the same vertex?
解析
只需考虑三类状态：
- ：三点各一只（初始状态）
- ：两只在同一顶点、另一只在另一个顶点
- 吸收态：三只同一顶点
设 为从 到吸收态的期望时间， 为从 到吸收态的期望时间。
从 出发：下一分钟若三只同向跳（顺时针或逆时针），仍在 ，概率 ；否则进入 ，概率 。故
从 出发：下一分钟
- 以概率 三只都跳到空顶点，直接吸收
- 以概率 变为
- 以概率 仍为
因此
解得 ，因此