The following is taken from a textbook I have called The History of Mathematics: A Brief Course by Roger Cooke. Note that I could not find an online reference to the actual mathematicians referred to in the problem so I have no idea what he’s referring to there. It is nonetheless immaterial for the problem.
A famous example of mathematical blunders committed by mathematicians (not statisticians, however) occurred some two decades ago. At the time, a very popular television show in the United States was called Let’s Make a Deal. On that show, the contestant was often offered the chance to keep his or her current winnings, or to trade them for a chance to win some unknown prize. In the case in question the contestant had chosen one of three boxes, knowing that only one of them contained a prize of any value, but not the contents of any of them. For ease of exposition, let us call the boxes, A, B, and C, and assume that the contestant chose box A.
The emcee of the program was about to offer the contestant a chance to trade for another prize, but in order to make the show more interesting, he had box B opened, in order to show that it was empty. Keep in mind that the emcee knew where the prize was and would not have opened box B if the prize had been there. Just as the emcee was about to offer a new deal, the contestant asked to exchange the chosen box (A) for the unopened box (C) on stage. The problem posed to the reader is: Was this a good strategy? To decide, analyze 300 hypothetical games, in which the prize is in box A in 100 cases, in box B in 100 cases (in these cases, the emcee will open box C instead to show it is empty), and box C in 100 cases. First, assume that in all 300 games the contestant retains box A. Then assume that in all 300 games the contestant exchanges box A for the unopened box onstage (either B or C). By which strategy does the contestant win more games?