We have an 11$\times$11 table of squares (consist of 121 squares of dimension 1$\times$1). we have 3 tiles shown in the picture. Each tile has dimension 1$\times$1. we now randomly pick 3 tiles into the table. Let $N$ denote the total of number of circles we get in a random drawing. Compute expected value of $N$, mean of $N$ and standard deviation of $N$.
I think of an idea that we construct a $X$-$Y$ coordinate in accordance with the 11$\times$11 table, the only way to get a circle is to put the first tile in square $(x,y)$ and $(x+1,y-1)$ and put the third tile in square $(x+1,y+1)$ and $(x,y-1)$ for $0 \le x,y \le 11$. I want to construct a variable that takes on value 1 if this arrangement occurs and 0 if not, then construct what I want. But now I get stuck. Any help would be really appreciated. thanks
The image of the three tile patterns is below