Expected Area inside Hexagon

Geometry Level 4

A regular hexagon with side length \(1\) is drawn. We flip \(6\) coins, with each coin corresponding to a unique midpoint of the hexagon. Then, we take all the midpoints of which their corresponding coin resulted heads, and connect them in clockwise order to create a polygon inside the hexagon. If the expected value of the area of this polygon can be expressed as \[\dfrac{a\sqrt{b}}{c}\] where \(a,b,c\) are positive integers, \(a,c\) are coprime and \(b\) is square-free, then find \(a+b+c\).

\(\text{Details and Assumptions}\)

A polygon with zero, one or two vertices has an area of \(0\).

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