paradox · traps
Bertrand paradox
Three fair recipes give three different answers.
Three fair ways to draw a chord. Do they give the same answer?
A long chord, drawn in green, is longer than a side of the yellow triangle. A short chord is red.
Lock a share, then draw a chord.
—This chord
—Chords that were long
—Long-run share for this recipe
The question is how often a random chord is longer than the side of an equilateral triangle inscribed in the circle.
Each recipe below is a fair way to say “pick a chord.” They do not pick from the same pile.
Two random ends are long about 1 in 3 times. A spot along a radius is long about 1 in 2. A midpoint chosen anywhere in the circle is long about 1 in 4.