A triangle and a rectangle are in semicircles of radius where two of their vertices are on the diameter, and the rests are on the arc. The angle of vertex of the triangle is equally probable to be any angle between and , while for a rectangle the height is equally probable to be any height between and . If you throw a dart and hit a polygon in the semicircle you will win a prize, but you are blindfolded, and you just have one chance. Which polygon will you choose? Why?
Assume that the dart has an equal chance of landing anywhere on the semicircle.