Psychologists Don't Know Math

stupefaction writes "The New York Times reports that an economist has exposed a mathematical fallacy at the heart of the experimental backing for the psychological theory of cognitive dissonance. The mistake is the same one that mathematicians both amateur and professional have made over the Monty Hall problem. From the article: "Like Monty Hall's choice of which door to open to reveal a goat, the monkey's choice of red over blue discloses information that changes the odds." The reporter John Tierney invites readers to comment on the goats-and-car paradox as well as on three other probabilistic brain-teasers."

Comments

Re:Ummm, I don't get it.

[Jeremy Erwin]

It's quite simple.

Suppose the car is behind door number one.

If you pick door number one, then Monty has a choice of picking door number two, or three. If you switch, you lose.

If you pick door number two, then Monty must open door number three. If you switch, you win.

If you pick door number three, then Monty must open door number two. If you switch, you win.

Monty's choice of which door to open is constrained in two out of three choices. Pick the door he didn't open, and you'll win two out of three times.

But the problem assumes that Monty has to offer you that choice. On the game show, he didn't.

Re:Hmmm....

[fractoid]

So the expected value of switching envelopes is 50% (0.5X + 2X), or 1.25X. This is wrong. If one envelope contains X and the other contains 2X then the expected gain G from switching is:
G = 50% * (Gained if we were holding X) + 50% * (Gained if we were holding 2X)
= 0.5 * (2X - X) + 0.5 * (X - 2X)
= 0

So switching envelopes doesn't change the expected value.