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Random connected area

Published on Saturday, 8th February 2020, 04:00 pm; Solved by 167

Problem 701

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.

3 random connected area

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

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