Expected Value (Tiles and Circles) 
I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: 
  Image
When I arrange the tiles in a $9\times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form?

Suppose we have a vertex (i.e. the center of the circle), each tile adjacent to it has a $\frac{1}{4}$ chance of 'turning towards' it. Since there're $4$ tiles, we have a $(\frac{1}{4})^{4}=\frac{1}{256}$ probability of forming a circle.
But we need a expected value, not the probability. I can't multiply by $99$ as that's incorrect. 
 A: Solving this question requires understanding the notion of linearity of expectation; let's say that the event $A_i$ for a given intersection $i$ in your grid is "A circle formed around the center $i$". We make a function called an indicator variable $\mathbb{1}_{A_i}$ that takes in a board and spits out 1 if there is a circle around center $i$ and 0 otherwise.
Now, expected number of circles is $\mathbb{E} \sum_{\textrm{centers } i} \mathbb{1}_{A_i}$, which by the linearity of expectations reduces to $\sum_{\textrm{centers }i} \mathbb{E}\mathbb{1}_{A_i}$. Now, the expected times that a given center will have a circle around it is precisely the probability that all of the circles line up, which is the probability you computed in the questions, so that sum now reduces to $\sum_{\textrm{centers }i} \frac{1}{256}$, i.e. you now have to count the number of possible centers. How many are there? (It is not 99; think of a 2x2 grid, there is only 1 center, in 3x2 grid there are 2, in 3x3 there are 4, in 4x3 there are 6; if you draw the grids, you can see what the formula is) 
