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Consider an m×n zero-one matrix A. An s×t submatrix of A is said to be even if the sum of its entries is even. In this paper, we focus on the case m=n and s=t=2. The maximum number M(n) of even 2×2 submatrices of A is clearly and corresponds to the matrix A having, e.g., all ones (or zeros). A more interesting question, motivated by Turán numbers and Hadamard matrices, is that of the minimum number m(n) of such matrices. It has recently been shown that for some constant B. In this paper we show that if the matrix A=A n is considered to be induced by an infinite zero one matrix obtained at random, then where E n denotes the number of even 2×2 submatrices of A n . Results such as these provide us with specific information about the tightness of the concentration of E n around its expected value of