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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?

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This is the third very similar question you post today without any personal input, whose solution is a direct application of the linearity of expectation. Let me at least hope that you begin to discern a pattern... – Did Aug 22 '14 at 13:55
up vote 15 down vote accepted

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.

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Thanks, this helped a lot! – heyhuehei Aug 22 '14 at 2:28
But if a circle is formed at a given vertex, the neihgbouring vertices can not have circles, so the events are not independent. – TonioElGringo Aug 22 '14 at 8:53
@TonioElGringo: True, but expected values of dependent events are still additive. – Ilmari Karonen Aug 22 '14 at 9:06

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 $E[X] = E[\sum X_i] = \sum E[X_i]= 8\cdot 10 \cdot (\frac14)^4$

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