Amanda Knox and bad maths in court
As an Italian court prepares to try Amanda Knox and Raffaele Sollecito for a second time on charges of killing Meredith Kercher, an expert says a judge failed to grasp the maths of probability involved in the case - and that courts often struggle when it comes to statistics.
British exchange student Meredith Kercher was found stabbed to death at the house she shared in Perugia, Italy, in November 2007.
Kercher's housemate, US student Amanda Knox, and the American's boyfriend Raffaele Sollecito, were accused of her murder, along with a third man, Ivorian drifter Rudy Guede.
Guede was convicted and jailed for 16 years. Knox and Sollecito were convicted of murder and sexual violence in December 2009 and jailed for 26 years and 25 years respectively. But in October 2011, the pair were freed on appeal after doubts were raised about the forensic evidence against them.
Then, last month, Italy's highest court overturned this acquittal, so Knox and Sollecito face a re-run of the appeal.
One of the key pieces of forensic evidence that helped to convict the pair in the first place was a kitchen knife found in Sollecito's kitchen, which was said to have Kercher's DNA on the blade.
But the DNA sample was tiny, and the appeal judge thought the evidence was unreliable, so he rejected a forensic scientist's suggestion to have it tested again.
"The sum of the two results, both unreliable… cannot give a reliable result," he wrote.
This was a mistake, according to mathematician Coralie Colmez.
"The thing with statistics and probability is that people feel it's intuitive," she says.
"But actually the mathematics tells you that if you have an unreliable test, you do it again and you can be a lot more certain of that answer. And it's not intuitive but a simple calculation will tell you that it's true."
Why are two tests better than one?
You have to view the two forensic tests as not separate from one another, but as one big test, says Colmez. She compares it to an experiment to find out whether a coin is fair or biased. The aim is to work out if the coin has been weighted to give heads 70% of the time by flipping it and looking at the results. You know there's a 50% chance it's fair, and 50% that it's biased.
"You do a first test and obtain nine heads and one tail... The probability that the coin is fair given this outcome is about 8%, [and the probability] that it is biased, about 92%. Pretty convincing, but not enough to convict your coin of being biased beyond a reasonable doubt," Colmez says.
"You do a second test, and this time you throw eight heads and two tails. Now the probability for a fair coin is about 16%, for a biased coin about 84%."
"So the naive thought might be that you haven't gained any certainty from this second test.
"But if you think about it differently, what you've really done is throw the coin 20 times and get 17 heads and three tails." This means there's a probability of 98.5% that the coin is biased.
"So what this means in the case of the knife in the murder is that if it were tested again, and once again the DNA was Meredith's profile we could be a lot more certain that the DNA on the knife is indeed Meredith's," Colmez says.
And if the knife were tested again and the DNA did not match Meredith Kercher's profile? That would be good news for Knox and Sollecito, she says.
"This would mean that this major piece of evidence against them would be discredited."
In Colmez's view this isn't a one-off. She says numbers get used and abused in court rooms all the time, and more use should be made of proper mathematical and statistics experts.
Another recent example she draws attention to is that of a Dutch nurse, Lucia de Berk, who was first arrested in 2001 after the death of a baby in her care at a hospital in The Hague, apparently from poisoning.
Afterwards, investigators found what they thought was a trend of suspicious deaths among 13 patients treated by De Berk in the previous four years. Five others almost died in what investigators said were suspicious circumstances.
In 2003, she was convicted of four murders and three attempted murders, and sentenced to life in prison.
Part of the evidence against her was the testimony of a statistician, who said the odds were 342 million-to-one that it was a coincidence she had been on duty when all the incidents occurred.
In 2004, an appeals court convicted her of three additional counts of murder and upheld the life sentence.
And in prison she might have stayed, if it hadn't been for the amateur statistical sleuthing of a doctor called Metta de Noo-Derksen, the sister-in-law of one of the doctors at the hospital where De Berk had worked. She had become suspicious of the reasoning used in the case and, along with her brother Ton, began a campaign to prove there had been a miscarriage of justice.
De Berk had been accused of causing some deaths, which she had later managed to prove had occurred when she hadn't been present in the hospital.
"But these deaths were just forgotten about in the trial and never spoken about again. And they never recalculated the probabilities," Colmez says.
"What the brother and sister team did was they went through all the deaths she was accused of, struck off the ones she had proven she wasn't even there for, and they recalculated the probability."
The probability of her being present for all the unexplained deaths was still enough to raise questions but if you consider the number of nurses at work in the Netherlands, you'd expect to see some unusual-looking - but innocent - clusters of unexplained deaths on some of their watches.
"Out of some 250,000 nurses, one would expect a couple of hundred to be involved in a set of circumstances similar to those of Lucia," Colmez says. "It definitely wasn't proof that she was a murderer."
After six years in prison, Lucia de Berk was acquitted in April 2010.
"It's a horrible, horrible story," says Colmez. "But it's uplifting that it was just some members of the public who went through a lot of work - and freed her."
Coralie Colmez is co-author of Math on Trial: How numbers get used and abused in the courtroom