Problem 213

Flea Circus

A 30×30 grid of squares contains 900 fleas, initially one flea per square.
When a bell is rung, each flea jumps to an adjacent square at random (usually 4 possibilities, except for fleas on the edge of the grid or at the corners).

What is the expected number of unoccupied squares after 50 rings of the bell? Give your answer rounded to six decimal places.

跳蚤马戏团

在30×30的方形网格上共有900只跳蚤，初始状态下每个方格上有一只跳蚤。
每响起一次钟声，每只跳蚤随机跳向一个相邻的方格（一般有4种选择，除非跳蚤是在网格的边上或角上）。

在响起50次钟声后，没有跳蚤的方格的期望数目是多少？将你的答案四舍五入到六位小数。