There are $99$ identical square tiles, each with a quarter-circle drawn on it. When the tiles are randomly arranged in a $9$ by $11$ rectangle, what is the expected value of the number of full circles formed?
I assume tiles can be rotated, but can't be turned over (which would hide the quarter-circle).
Hint 1: A full circle is formed at any of the $80$ interior vertices if the four tiles touching that vertex all have the proper orientation. What is the probability of that?
Hint 2: Expected value of sum = sum of expected values.
lets match each formed circle to its upper left corner, there are 8x10 such upper left tiles possible, the probability for each of those tiles to form a circle:
$ P($tile $i$ formed a circle and is in the upper left corner$)= (\frac14)^4 $
(each tile of the 4 has a 1/4 chance of being in the right oriantation)
therefore $E[X] = E[\sum X_i] = \sum E[X_i]= 8\cdot 10 \cdot (\frac14)^4$