Bertrand paradox

The Bertrand paradox illustrates how different methods of randomly selecting a chord in a circle can lead to varying probabilities of the chord being longer than a side of an inscribed equilateral triangle. Essentially, the way you choose the chord—by fixing endpoints, picking a midpoint, or selecting a random radius—affects the likelihood of the chord having a certain length. This paradox highlights that "randomness" depends on the method used, and without a precise definition, probability outcomes can differ significantly. It's a classic example showing the importance of clarity in defining random processes in statistical problems.