But the failure of expected value to capture human intuitions becomes clear when you ask yourself why you'd be reluctant to pay even a measly $1,000 for the privilege of playing this game. That measly $1,000 is of more utility to you than are the billions of dollars that are only a remote possibility.
(Similar sorts of declining utility characterize other sorts of wealth. Someone who's publicity-rich will get a much smaller kick out of an article by him or her in a magazine, say, than someone who's never been published before. Someone who's traveled extensively all over the world will get less out of a few days in Florence than will someone else who's never left Torrance. And someone who's had sex with many partners will get … Well, you get the idea.)
To further discombobulate yourself, consider the so-called Ellsberg paradox, named after Daniel Ellsberg of Pentagon Papers fame. Imagine a large urn -- mathematicians like large urns almost as much as they like coins and dice -- containing 300 marbles, exactly 100 of them red, and the other 200 of them black and yellow in unknown proportions (i.e., from 0 to 200 black, the others yellow).
You're asked to choose option A or option B. Under A, you'll receive $1,000 if you pick a red marble from the urn, whereas under B, you'll receive $1,000 if you pick a black marble. Which option would you take?
You're also asked to choose between option C and option D. Under C, you'll receive $1,000 if you pick a red or a yellow marble from the urn, whereas under D you'll receive $1,000 if you pick a black or a yellow marble from the urn. Which option would you take here?
You'll prefer option A to option B exactly when you think picking a red marble from the urn is more probable than picking a black marble. Likewise, you'll prefer option C to option D exactly when you think picking a red or yellow marble from the urn is more probable than picking a black or yellow marble.
It would seem too that if you prefer option A to option B, you should also prefer C to D. And, if your prefer option D to option C, you should also prefer B to A.
The only problem is that this is not what people actually do. Most prefer Option A to B (presumably because they know for sure that there are 100 red marbles in the urn), but they prefer option D to option C (presumably because they know for sure that there are 200 black and yellow marbles in the urn). People are averse to uncertainty and choose the sure thing even when their behavior is inconsistent and violates the usual axioms of utility theory and subjective probability.
Most of us don't like risk and uncertainty. That's too bad, because there's no shortage of either.
-- Professor of mathematics at Temple University, John Allen Paulos is the author of best-selling books, including "Innumeracy" and "A Mathematician Plays the Stock Market." His "Who's Counting?" column on ABCNews.com appears the first weekend of every month.