Given an infinite rectangular grid, each square is colored independently with probability $p$. Adjecent colored squares form blobs. The size of a blob is the number of sqaures that make it up.
Q1.
If we look at all the blobs, what is the expected value of the size (as a function of $p$)? If there is no closed formula, can you at least give me asymptotic estimates near $p=0$ and $p=1$? It obviously tends to $\infty$ as $p\to1$, but how quickly?
Q2.
We fix $N\in\mathbb{N}$, and consider only blobs of size $N$. They can have a finite number of possible shapes. Does the distribution over these shapes depend on $p$? (Ie. do certain shapes become more or less likely if $p$ is changed?)