## 2×2 Squares on a Randomly Filled m×n Board

June 28, 2010 at 17:03 (algorithm, java, math.PR)

This post, like the last one, is an offshoot of a thread on Math Help Forum. The situation is: Suppose there is an m×n chess-like board and we wish to fill in all the (1×1) squares according to: all squares are either black or white, and the probability that a square is black is $p$. What is the probability that a randomly filled board will contain at least one 2×2 black square?