KöMaL 2012 B4451

Consider a convex $n$-sided polygon that has no four vertices lying on the same circle. Three vertices are selected in every possible way, and the circle passing through the three points is drawn. Such a circle is called thin if the vertices of the polygon (except the vertices of the triangle) all lie outside the circle. The circle is called fat if all vertices of the polygon lie on the closed disc bounded by the circle. Are there more thin circles or more fat circles?