Special Reports

YOU are serving on a jury in the trial of a man accused of murdering his wife. It emerges that the accused had regularly beaten his wife—but his highly paid defence team presents statistics supposedly showing that only 1 in 1000 wife beaters go on to kill their wives. The conclusion seems obvious: you should not make much of the wife-beating evidence.

Wrong. You have just fallen into a trap that awaits anyone who relies on "common sense" to understand evidence based on probabilities. For that 1 in 1000 figure is a red herring. You should be focusing not on the odds of wife beaters going on to kill their wives, but the odds that a wife beater whose wife has been murdered is responsible for her death.

Confused? You are not alone. It is a trap so subtle that neither the defence nor the prosecution spotted it during the O. J. Simpson trial, from which these figures are taken. It took a professor of statistics, Jack Good at the Virginia Polytechnic Institute, to bring the error to light in the letters pages of the journal Nature (vol 375, p 541).

Using probability theory, Good showed that the chances of a wife beater being guilty of his wife's murder depends on more than just the proportion of wife beaters who progress to murder. It also depends on the chances of her being murdered at all, and the chances that in any given year a husband will kill his wife. When Good combined these probabilities, he found that the chances of a wife beater being guilty of his wife's murder were around 50:50.

What difference Good's subtle calculations would have made to the outcome of the O. J. Simpson case will never be known: they were never presented. In Britain, following a bizarre ruling from the Appeal Court last month, juries are specifically forbidden access to such calculations. To the astonishment of many legal experts, the court upheld a previous provisional ruling that "formulas, mathematical or otherwise" are "inappropriate" for juries—common sense can be relied on instead.

The experts say the decision can only boost the risk of miscarriages of justice. Colin Howson, professor of logic at the London School of Economics says the ruling is astonishing. "It would be funny, if people's freedom did not depend on it."

The ruling came at the end of one of the most vexed legal cases of recent years—one that showed just how difficult it is for juries to interpret probabilities. In 1991, a 24-year-old woman was raped in Hemel Hempstead, Hertfordshire. Police arrested a local man, Dennis Adams. DNA tests appeared to point strongly towards Adams's guilt, and according to the forensic scientists the odds of getting so good a DNA match from someone innocent of the rape were just 1 in 200 million.

All the other evidence in the case, however, pointed to Adams's innocence: his alibi was never undermined by the prosecution, and his victim not only failed to pick him out of an identity parade but even admitted in court that he did not look like her attacker.

During the trial, Adams's lawyers warned the jury that the lack of supporting evidence greatly diluted the strength of the DNA evidence. They pointed out that the 1 in 200 million was not the chance of Adams being innocent, but was instead an estimate of how much one's initial belief in his guilt should be amplified by the forensic evidence. And if the chances of guilt suggested by the other evidence were very low, then even multiplying them by a factor of many millions could still leave grounds for reasonable doubt.

But to no avail: Adams was convicted and sentenced to seven years in prison. His lawyers appealed, arguing that the trial judge had failed to explain clearly the probability arguments in the case. In May last year, the Appeal Court upheld the appeal and ordered a retrial—but the appeal judges then stunned the defence team by stating that the trial judge had spent too much time explaining the scientific approach to assessing such evidence. The appeal judges also argued that such techniques were "too rigid" and led juries into "inappropriate and unnecessary realms of theory and complexity" (This Week, 8 June 1996, p 7).

At his retrial last December, Adams was again found guilty, and the defence again lodged an appeal. That was rejected in October—and the Appeal Court has now confirmed its original ruling. The laws of probability, it has declared, are "a recipe for confusion, misunderstanding and misjudgment". Instead, juries should rely on their "individual common sense and knowledge of the world".

But relying on common sense can lead juries into a minefield of fallacies, says David Balding, a professor of statistics at Reading University who has served as an expert witness in DNA trials. "Studies have shown that people cannot reason very well at all when faced with probabilities," he says.

Such concerns are not academic nit-picking—as recent miscarriages of justice show. In August 1996, the Appeal Court quashed the conviction of Alan Doheny, who was sentenced to 12 years in 1990 for rape and other offences largely on the basis of DNA evidence. The Appeal Court judges said that the Home Office's key forensic scientist had fallen for the "prosecutor's fallacy"—just as Balding and others had warned would happen ("Improving the odds on justice", 16 April 1994, p 12).

The prosecutor's fallacy occurs when the prosecution interprets statistics about DNA and blood group evidence in the wrong way. For example, a so-called match probability of 1 in 40 million represents the chances of getting so good a match assuming that the person is innocent. What the jury is trying to decide, however, is something different: the probability of innocence, given the DNA information.

Yet in presenting DNA and blood group evidence in the Doheny case, the Home Office scientist misled the jury, implying that the 1 in 40 million match probability was the chances of Doheny being innocent—and the jury duly convicted him.

According to Peter Donnelly, a professor of statistics at Oxford University who has served as an expert witness in many cases involving probabilistic evidence, the Doheny case is the fourth to feature the prosecutor's fallacy and the third in which the appeal has been successful. "The question is, how many other cases have there been where the same mistake was made?"

The Forensic Science Service, the British government's executive agency charged with providing prosecution evidence in courts, tells New Scientist that it is aware of the dangers of the prosecutor's fallacy. "We are alert to it," a spokesman says.

But with the new Appeal Court ruling, juries have been left wide open to a raft of even more subtle traps. Balding says: "Explaining probability theory in court may cause confusion, but it is likely to cause less harm than not explaining it."