time, were not convinced. On June 11, 2003, at the end of the six-and-a-half-week trial, they found her not guilty on all three counts of murder. The jury had decided that the deaths of the three babies, as in the case of Sally Clark, had been a tragic coincidence. Whatever the odds against something happening, the fact that odds can be calculated means that it can, and, given enough time, will happen.
Odds of 73 million to 1, although inaccurately applied in the case of Sally Clark, will, eventually, come up. Even if those had, in fact, been the odds against the deaths of her three children being coincidence, it would not have pointed unerringly to her guilt. A 73 million to 1 chance occurrence isnât an unimaginable likelihood. If one in every seventy-three million people were green, then thereâd be eighty-four green people in the world. Shouldnât be too hard to spot.
Judges and juries are regularly asked to weigh up odds of 1 in 3 millionâin cases where DNA samples are presented as crucial evidence.
And, of course, they get it wrong.
In 1990, Andrew Deen was sentenced to sixteen years in jail for raping three women. The main evidence linking Deen to the attacks was the close match between DNA samples found at the scene of the crimes and those from Deen. At the trial, the forensic scientist presenting the DNA evidence said that the match was so good that the probability of the samples having come from someone other than Deen was 1 in 3 million. In his summing up, the judge told the jury that so large a figure, if correct, âapproximates pretty well to certainty.â There could be no coincidence.
But on appeal the court quashed the conviction, declaring the verdict unsafe. It decided that both the forensic scientist and the judge had fallen into a trap known as the âprosecutorâs fallacy.â They had assumed that the DNA evidence meant that there was only a 3-million-to-1 chance that Deen was not guilty. But they were mistaken.
For the true picture, the appeal court judges turned to a mathematical theorem constructed by a nineteenth-century cleric. Bayesâ Theorem addresses the laws of âinverse probability.â It provides a formula for working out the impact of new evidence (like DNA samples) on the existing odds of guilt or innocence prior to the introduction of the new evidence.
If âprior probabilityâ of guilt is smallâif there is little other evidence to corroborate the DNA evidenceâthen even the impressive probabilities of genetic fingerprinting can be dramatically diminished.
Researchers at the Institute of Environmental Health and Forensic Sciences in Auckland, New Zealand, use crime statistics and âBayesian reasoningâ to estimate typical prior probabilities. They found that even a DNA match with odds of millions to one can be cut down to final odds against innocence of just 3 to 1âleaving plenty of room for âreasonable doubt.â
So if you are currently stuck in an intractable legal dispute over the probability of something or other having happened, or not happened, as the result of pure coincidenceâhelp is at hand. Try applying Bayesâs handy mathematical formula.
Good luck.
6
LUCK OR COINCIDENCE?
Itâs the day of the Kentucky Derby and you grudgingly hand over your hard-earned $20 for the office pool. Your horse, which has begun with a moderate chance of success, mysteriously chooses to stop half way around the course to admire the stamina and athleticism of its four-legged friends. Your colleague George Robertson wins the jackpot. His horse, a rank outsider, confounds the bookiesâ pessimistic expectations. This is the seventh time George has won the sweepstake in ten years.
Do you say, âWell done George, itâs good to see that the rules of probability are still functioning and that your chances of winning this year were not materially diminished by the fact that you have won so many
Odds of 73 million to 1, although inaccurately applied in the case of Sally Clark, will, eventually, come up. Even if those had, in fact, been the odds against the deaths of her three children being coincidence, it would not have pointed unerringly to her guilt. A 73 million to 1 chance occurrence isnât an unimaginable likelihood. If one in every seventy-three million people were green, then thereâd be eighty-four green people in the world. Shouldnât be too hard to spot.
Judges and juries are regularly asked to weigh up odds of 1 in 3 millionâin cases where DNA samples are presented as crucial evidence.
And, of course, they get it wrong.
In 1990, Andrew Deen was sentenced to sixteen years in jail for raping three women. The main evidence linking Deen to the attacks was the close match between DNA samples found at the scene of the crimes and those from Deen. At the trial, the forensic scientist presenting the DNA evidence said that the match was so good that the probability of the samples having come from someone other than Deen was 1 in 3 million. In his summing up, the judge told the jury that so large a figure, if correct, âapproximates pretty well to certainty.â There could be no coincidence.
But on appeal the court quashed the conviction, declaring the verdict unsafe. It decided that both the forensic scientist and the judge had fallen into a trap known as the âprosecutorâs fallacy.â They had assumed that the DNA evidence meant that there was only a 3-million-to-1 chance that Deen was not guilty. But they were mistaken.
For the true picture, the appeal court judges turned to a mathematical theorem constructed by a nineteenth-century cleric. Bayesâ Theorem addresses the laws of âinverse probability.â It provides a formula for working out the impact of new evidence (like DNA samples) on the existing odds of guilt or innocence prior to the introduction of the new evidence.
If âprior probabilityâ of guilt is smallâif there is little other evidence to corroborate the DNA evidenceâthen even the impressive probabilities of genetic fingerprinting can be dramatically diminished.
Researchers at the Institute of Environmental Health and Forensic Sciences in Auckland, New Zealand, use crime statistics and âBayesian reasoningâ to estimate typical prior probabilities. They found that even a DNA match with odds of millions to one can be cut down to final odds against innocence of just 3 to 1âleaving plenty of room for âreasonable doubt.â
So if you are currently stuck in an intractable legal dispute over the probability of something or other having happened, or not happened, as the result of pure coincidenceâhelp is at hand. Try applying Bayesâs handy mathematical formula.
Good luck.
6
LUCK OR COINCIDENCE?
Itâs the day of the Kentucky Derby and you grudgingly hand over your hard-earned $20 for the office pool. Your horse, which has begun with a moderate chance of success, mysteriously chooses to stop half way around the course to admire the stamina and athleticism of its four-legged friends. Your colleague George Robertson wins the jackpot. His horse, a rank outsider, confounds the bookiesâ pessimistic expectations. This is the seventh time George has won the sweepstake in ten years.
Do you say, âWell done George, itâs good to see that the rules of probability are still functioning and that your chances of winning this year were not materially diminished by the fact that you have won so many