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Problem 701

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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.

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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。

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