Problem 701

Random connected area

Consider a rectangle made up of $W \times H$ square cells each with area $1$ .
Each cell is independently coloured black with probability $0.5$ otherwise white. Black cells sharing an edge are assumed to be connected.
Consider the maximum area of connected cells.

Define $E (W, H)$ to be the expected value of this maximum area. For example, $E (2, 2) = 1.875$ , as illustrated below.

You are also given $E (4, 4) = 5.76487732$ , rounded to $8$ decimal places.

Find $E (7, 7)$ , rounded to $8$ decimal places.

随机连通区域

考虑由 $W \times H$ 个面积为 $1$ 的正方形方格组成的长方形。
每一格正方形独立地以 $0.5$ 的概率染上黑色或白色。如果两个黑色正方形格共用一条边，则认为它们是连通的。
考虑最大连通区域的面积。

记 $E (W, H)$ 为最大连通区域的面积的期望值。例如， $E (2, 2) = 1.875$ ，如下图所示（其中数字表示最大连通区域的面积）。

已知 $E (4, 4) = 5.76487732$ ，保留 $8$ 位小数。

求 $E (7, 7)$ 并保留 $8$ 位小数。