Sunday, June 08, 2008

Behavioral Probability Errors

The New York Times has a review of a book on how unintuitive probability is. The Drunkard's Walk by Leonard Mlodinow goes over many examples where average, and really smart people, get it wrong (listen to interview here, and see Random House's book excerpt here). For example, The famous Monte Hall problem, from Let's Make a Deals, presented contestants with three doors, behind one of which is a new car. You take your pick, but before your fate is revealed, the M.C. swings open one of the other doors, revealing a booby prize (a goat). So far, so good, but now comes the big decision. Do you stay with your original choice or switch to the other unopened door? Many people say the odds don't change, because there will always be a goat, so the odds have to be 50-50 (two doors left, one car).

To see this, consider the following cases behind doors 1, 2 and 3 respectively: goat, goat, car. If you pick one of the first two, you will lose if you don't switch. So, switching changes your odds from 0 to 100%, because revealing the goat leaves the car. If you pick the third choice, switching takes you from a 100% chance of winning, to a 0% chance. 1/3(0+0+100%)=1/3 are your original odds. Your new odds are 1/3(100%+100%+0%), or 2/3.

This has been studied a lot (over 40 academic papers), but what is interesting that many people got this wrong, even very smart people like Paul Erdos.

He also goes over the meaning of a Positive HIV test that is 99.9% accurate. Using bayesian statistics, the actual odds of a positive test are less than 10%. And then there's the OJ trial, where one of Simpson’s lawyers, Alan Dershowitz, noted that even though OJ beat his wife, that hardly mattered, because in the United States, four million women are battered every year by their male partners, yet only one in 2,500 is ultimately murdered by her partner (1 in 1000), so, by the 'reasonable doubt' criterion, this is irrelevant.

The jury may have found that persuasive, but it’s a spurious argument. The relevant question was what percentage of all battered women who are murdered are killed by their abusers, which isn't 1 in 1000, but rather 9 in 10.

I think the main thing brought up by all this is that humans have very poor intuition of conditional probability. Yet it must be remembered that it is still very difficult to make money, implying that, like an options pricing model, when the economic implications are large, people override their intuition with models. I don't think these biases have much meaning for asset pricing, because the stakes are too large for the errors to remain, and so, I don't find them so important

The second main take away is that really smart people are often wrong, on questions of pure logic. No matter how smart we are, and Erdos and Dershowitz are smart, we aren't always right.


Anonymous said...

My guess is Dershowitz was more concerned with being effective than he was with being right.

Michael Stack said...

Second the Dershowitz comment. In what context did he make the argument? If it's an argument he made outside court after a decision was rendered then I'd be more willing to believe that he's mistaken. Otherwise, I suspect he understood exactly what he was doing.

Anonymous said...

Sometimes I'm amazed that the world is as civilized as it is, when truckers in G7 countries blockade roads because gas prices are high.