Pick a 'random' chord in a circle. The odds change depending on how you pick.

It depends entirely on your method of picking 'randomly' — that's the paradox. Fix two random points on the circle's edge and join them: 1/3. Fix a random radius and a random point along it, then draw the chord perpendicular there: 1/2. Fix a random point anywhere inside the circle as the chord's midpoint: 1/4. Joseph Bertrand posed this in 1889 to show 'at random' is meaningless without a precise selection process.