One of the legal journals I subscribe to has a piece on "cognitive illusions" that starts out with the Monty Hall "paradox":
You are a contestant on a game show with three doors. Behind one door is $10,000 cash; behind the other two doors are goats. The host asks you to choose the door you want, and you select door number one. The host opens door number three, revealing a goat, and then gives you the opportunity to keep what's behind door number one or switch to door number two. What do you do?
Most attorneys [and some mathematicians -- AC] would answer that the decision makes no difference, because there is a 50-50 chance that the cash is behind door number one. That answer is wrong. From the moment you chose door number one, you had a one-third chance of having selected the door with the cash behind it. The other two doors, combined, had a two-thirds chance of hiding the cash. When it is revealed that door number three does not have the cash, then door number two, alone, has a two-thirds probability of hiding the cash. Thus, when given the choice, you should always switch doors, because you are twice as likely to win the game.
The writer then says the same principle applies to "Deal or No Deal." The contestant picks one of 26 briefcases. One of the 26 has $1 million in it. "As more and more unpicked briefcases are opened without revealing the $1 million, the contestant believes that the odds he or she picked the right briefcase at the start must be increasing, resulting in the rejection of increasingly high-dollar 'deals.' In fact, the exact opposite is true: the odds are increasing -- [up to 25/26] -- that one of the unpicked and unopened briefcases actually contains the $1 million, and the contestant will be left with very little."
This particular cognitive illusion is indeed deceptive because I think the author is wrong. He's not wrong about the Monty Hall paradox. That's right, but he leaves out the critical assumption, which is what trips up most people the first time they hear this: For the sake of suspense, Monty Hall never opens the door with the money.
The author is wrong about "Deal or No Deal." Contestants on "Deal or No Deal" pick briefcases at random and (almost always) hit the $1 million. Suppose a contestant gets down to two briefcases with the $1 million still on the board. Does he have a 25/26 chance of winning the million if he trades? If Howie had been picking briefcases, and we knew that Howie would never pick the $1 million briefcase, then yes. If the contestant had been picking at random, then no.
Update: I've done a more rigorous analysis using Bayes' theorem.
