In weak imitation of Archimedes, let’s talk mathematics in the time of war. I had some formal training in the subject but lately I’ve grown fond of popular books explaining basic concepts of math and science. A good explanation can be of great educational value – I’m always looking for better ways to explain math my own child – and, besides, I am normally able to check whether the author is being rigorous enough.

Years ago, I bought The Drunkard’s Walk by Leonard Mlodinow, probably at a US airport bookstore, but only got down to reading it last December. Mlodinow is a physicist and a co-author of Stephen Hawking’s, an accomplished scientist and an effective writer.

The subject matter of The Drunkard’s Walk is likely unoriginal. The history of probability and statistics part is textbook stuff. Bayesian logic, the Gaussian distribution, the central limit theorem and the numerous fallacies have long been known to students of the field. But as I’ve just said, I believe that every book that explains at least one important and/or fine point better than its predecessors deserves to be printed. A popular book that explains a wider range of topics in a manner accessible to the general public should also be welcomed. Mlodinow’s book surely gave me plenty to think about.

Here’s what Mlodinow writes about a relatively minor but interesting episode from the O.J. Simpson trial. It concerns Alan Dershowitz, the man alternatively called “a highly talented and honorable advocate, and a fearless public intellectual” and “the obsequious loophole-artist for such fragrant clients as Claus von Bulow and OJ Simpson,” and Jeffrey Epstein more recently.

The prosecution made a decision to focus the opening of its case on O.J.’s propensity toward violence against Nicole. Prosecution spent the first ten days of the trial entering evidence of his history of abusing her and claimed that this alone was a good reason to suspect him of her murder. As they put it, “a slap is a prelude to homicide.” The defense attorneys used this strategy as a launch pad for their accusations of duplicity, arguing that the prosecution had spent two weeks trying to mislead the jury and that the evidence that O.J. had battered Nicole on previous occasions meant nothing. Here is Dershowitz’s reasoning: 4 million women are battered annually by husbands and boyfriends in the United States, yet in 1992, according to the FBI Uniform Crime Reports, a total of 1,432, or 1 in 2,500, were killed by their husbands or boyfriends. Therefore, the defense retorted, few men who slap or beat their domestic partners go on to murder them. True? Yes. Convincing? Yes. Relevant? No. The relevant number is not the probability that a man who batters his wife will go on to kill her (1 in 2,500) but rather the probability that a battered wife who was murdered was murdered by her abuser. According to the Uniform Crime Reports for the United States and its Possessions in 1993, the probability Dershowitz (or the prosecution) should have reported was this one: of all the battered women murdered in the United States in 1993, some 90 percent were killed by their abuser. That statistic was not mentioned at the trial.

From what I have since read, Dershowitz appears to have made that argument not directly to the jury but to the judge, in a brief prepared for a preliminary hearing. Later, however, he repeated it on TV and in his book about the trial. He argued that the evidence of O.J. beating his wife should be excluded because it could easily prejudice the jury. He claimed, correctly but irrelevantly, that very few wife-beaters go on to become wife-killers. Because of that – according to Dershowitz – OJ’s wife-beating was merely character evidence, which is normally inadmissible at the trial stage. This logic could have worked if OJ’s ex-wife had been alive. But the fact that she had been murdered made that evidence statistically significant, as Mlodinow explains in the long passage above.

As early as 1995, I. G. Good of Virginia Tech (in Nature [pdf]), Jon Merz and Jonathan Caulkins (in Chance Magazine), and John Allen Paulos of Temple University (in The Philadelphia Inquirer) explained – in slightly divergent ways – how Dershowitz got that one wrong.

It’s also explained in this presentation [PDF] by prof. Richard Gill of Leiden University (not to be confused with the forensic DNA guru Peter Gill), who played a major role in the exoneration of Lucia de Berk.

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