Monty Hall and “the Leibniz Illusion”

摘要

Hardly any mathematical problem has been debated so fiercely and widely as the Monty Hall problem.The question of why so many people were mistaken, and moreover, why they were so deeply convinced that they were right and Marilyn vos Savant was wrong, still awaits a proper answer. Steven Pinker and other cognitive psychologists before him blame the equiprobability bias. I will show that the equiprobability bias is in fact a sound stochastic intuition. But through our labeling the three doors, a “probability shift” occurs. Instead of looking for the probability that we picked the door with the car behind it, we are now looking for the probability that the car is behind the door we picked. As a result, Marilyn’s critics suffer from a stochastic illusion that I have called “the Leibniz Illusion”. This illusion makes us believe that, after the host opens one of the three doors, there remain only two alternative possibilities. The uniformity belief now says that chances are fifty-fifty that the car is behind the door we picked. In reality, there still remain three alternative possibilities for the door we picked. This seems paradoxical, since only two doors are left. Once we compare our picking one of the three doors to drawing a playing card from a deck and realize that just like the playing cards each of the three doors has its own hidden label, the paradox is resolved and it becomes clear how Marilyn’s critics were deluded