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.

This problem cannot be solved as simply as a normal math problem, because it requires advanced mathematical understanding and computer science skills. The description indicates a problem of expected values in a randomly generated grid, which involves quite some randomness and complexity.

Roughly, we can consider the problem as a Markov process or dynamic programming problem, where each state represents a possible configuration of the colored grid. The transitions between states are defined by the black or white coloring of the squares, and the expected value is computed by observing the maximum area of connected black cells over multiple different random grid configurations.

However, directly calculating the expected value for a 7×7 grid is extremely time costing and computationally complex, as there are 2^(49) possible states of the grid, which is a huge number, making it practically impossible to directly calculate the expected value.

Therefore, we would need to develop an efficient algorithm to tackle this problem instead, which could involve techniques such as memoization and state compression to reduce the state space, or using Monte Carlo sampling methods to estimate the expected value. These advanced techniques are topics studied in computer science and probability theory.

Unfortunately, as a text-based AI, I am unable to implement such complex algorithms, and I recommend that you study these methods, which could help you to dive deeper into this problem. You may also find some online coding platforms or mathematical software where you can implement and run the algorithm to get the value of E(7, 7).

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