Paradox

PASCAL’S IMPROBABLE CALCULATIONS 225wins whatever prize is behind the door. Behind one door is avaluable prize while the other two contain prank prizes, suchas a goat. Suppose that you are a contestant and have pickeddoor number 1. To increase the suspense, Monty shows youwhat is behind door 2. You expect to see no prize behind thedoor Monty shows because you know that he always revealsa losing door. He then offers you the chance to trade yourclaim on what is behind door 1 for what is behind doornumber 3. Should you switch?In the September 1991 issue of the Parade magazinecolumn “Ask Marilyn,” Marilyn vos Savant advised switching.Many mathematicians scolded Marilyn for committinga fallacy.But Marilyn was right. Most people infer that switchingis pointless because you already know that Monty will reveala losing door. And indeed, the two remaining doors wouldhave an equal chance of being winners if Monty Hall hadrandomly revealed a door. However, Monty only revealslosing doors. Monty’s revelation that door 2 has a goat cannotraise the probability that door 1 has the prize because youalready knew that Monty was going to either reveal 2 as aloser or reveal 3 as a loser. However, Monty’s revelation thatdoor 2 is a loser does increase the probability that 3 is thewinner. Prior to this news, door 3 only had a 1/3 chance ofholding the prize. After learning that door 2 is a loser, theprobability that the prize is behind door 3 rises to 2/3 becausethe probability of door 1 winning is not affected.Nearly everybody gets the Monty Hall problem wrong.Many resist the correct answer even when it is explained tothem. Despite Monty’s extensive roster of victims, I amreluctant to describe “Should you switch doors?” as a paradox.True, there are conflicting good answers if brute persuasive-