Bertrand's box paradox

It was first posed by Joseph Bertrand in his 1889 work Calcul des Probabilités.

This simple but counterintuitive puzzle is used as a standard example in teaching probability theory.

The solution illustrates some basic principles, including the Kolmogorov axioms.

Instead, one should sum the probabilities that the cases would produce the observed result.

A survey of psychology freshmen taking an introductory probability course was conducted to assess their solutions to the similar three-card problem.

The paradox starts with three boxes, the contents of which are initially unknown
Bertrand's box paradox: the three equally probable outcomes after the first gold coin draw. The probability of drawing another gold coin from the same box is 0 in (a), and 1 in (b) and (c). Thus, the overall probability of drawing a gold coin in the second draw is ⁠ 0 / 3 ⁠ + ⁠ 1 / 3 ⁠ + ⁠ 1 / 3 ⁠ = ⁠ 2 / 3 ⁠ .