# Probability is not intuitive, ctd.

As further evidence for “Non-statisticians often don’t believe statisticians,” I submit to you a New York Times Book Review of “The Drunkard’s Walk” by Leonard Mlodinow.

The reviewer, George Johnson, starts by writing:

State lotteries, it’s sometimes said, are a tax on people who don’t understand mathematics. But there is no cause for anyone to feel smug. The brain, no matter how well schooled, is just plain bad at dealing with randomness and probability. Confronted with situations that require an intuitive grasp of the odds, even the best mathematicians and scientists can find themselves floundering.

Agreed. Johnson goes on to give two great examples of where failing to condition on all of your data can have life-changing consequences:

Hardest of all for our blinkered brains are cases involving Bayesian statistics, where one must gauge how the probability of one event hinges on that of another. Mlodinow learned the difficulty firsthand nearly 20 years ago when his doctor told him, out of the blue, that it was 99.9 percent certain that he was infected with H.I.V. Mlodinow had none of the risk factors (except for being human), but he had scored H.I.V.-positive on a test that had a false positive rate of one in 1,000.

If his doctor had studied probability in medical school, he would have seen the situation in a different light. Statistics from the Centers for Disease Control and Prevention showed that in Mlodinow’s demographic group, one in 10,000 people tested positive and was ultimately confirmed as carrying the virus. In addition, there were the statistical flukes — the 10 (one in 1,000) who were false positives. Compare those numbers, and the chance that Mlodinow was infected (he wasn’t) was one in 11.

When statistics are used in a court of law the effect can be just as misleading. Mlodinow recalls the O. J. Simpson trial, in which the prosecution depicted the defendant as an inveterate wife abuser. One of Simpson’s lawyers, Alan Dershowitz, countered with statistics: 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.

The jury may have found that persuasive, but it’s a spurious argument. Nicole Brown Simpson was already dead. The relevant question was what percentage of all battered women who are murdered are killed by their abusers. The answer, Mlodinow notes, didn’t come up in the trial. It was 90 percent.

That is good stuff. But what I remember two years later is this strange moment:

If a woman has two children and one is a girl, the chance that the other child is also female has to be 50-50, right? But it’s not. Cardano again: The possibilities are girl-girl, girl-boy and boy-girl. So the chance that both children are girls is 33 percent. Once we are told that one child is female, this extra information constrains the odds. (Even weirder, and I’m still not sure I believe this, the author demonstrates that the odds change again if we’re told that one of the girls is named Florida.)

Come on – “I’m still not sure I believe this”? Then maybe he should have read the passage where Mlodinow makes this argument a few more times. Mlodinow lays it out quite clearly, and the NYT has no excuse for insinuating that his statistical reasoning is anything less than compelling.