Further and deeper exploration of paradoxes and challenges of intuition and logic can be found in my recently published book, Probability, Choice and Reason.

On the 9th of November, 1999, Sally Clark, a 35-year-old solicitor and mother of a young child, was convicted of murdering her first two children. The presiding Judge, Mr. Justice Harrison, declared that “… we do not convict people in these courts on statistics. It would be a terrible day if that were so.” As it turned out, it was indeed a terrible day, for Sally Clark and for the justice system.
The background to the case is that the death of the babies was first put down to SIDS (‘Sudden Infant Death Syndrome’). Then a Home Office pathologist expressed doubts and Sally Clark was charged with murder. It later transpired that essential evidence in her favour had not been disclosed to the defence, but not before a failed appeal in 2000. At a second Appeal, in 2003, she was set free, and the case is now recognised as a classic miscarriage of justice.
So what went wrong?
A key turning point in the trial was the evidence given by a key prosecution witness, who argued that the probability of a baby dying of SIDS was 1 in 8,543. So the probability of two babies dying, he said, was that fraction squared, or 1 in about 73 million. But one of the basic laws of probability is that you can only multiply probabilities if those probabilities are independent of each other. That assumes no genetic, environmental or other innocent link between these sudden deaths at all, even assuming the 1/8543 number was correct. The other error is arguably more sinister, and very difficult for the layman to detect. It is known as the ‘Prosecutor’s Fallacy’.
The ‘Prosecutor’s Fallacy’ is to represent the probability of innocence given the available evidence to be the same as the probability of the evidence arising given the fact of innocence. In fact, they are very different. In particular, the following two propositions differ markedly.
1. The probability of observing some evidence (the dead children) given that a hypothesis is true (here that Sally Clark is guilty).
2. The probability that a hypothesis is true (here that Sally Clark is guilty) given that we observe some evidence (the dead children).
Notably, the probability of the former proposition is much lower than of the latter.
Indeed, the probability of the children dying given that Sally Clark is a child murderer is effectively 1 (100%). However, the probability that she is a child murderer given that the children have died is a whole different thing.
Critically, we need to consider what is known by statisticians as the prior probability that she would kill both babies, i.e. the probability that she would kill her children, before we are given this evidence of sudden death. This concept is central to what is known as Bayesian reasoning. Critically, this prior probability must not be viewed through the lens of the later emerging evidence. It must be established on its own merits and then merged through what is known as Bayes’ Rule with the emerging evidence.
In establishing this probability, we need to ask whether there was any other past indication or evidence to suggest that she was a child murderer, as the number of mothers who murder their children is close to vanishingly small. Without such evidence, the prior probability of guilt is close to zero. In order to update the probability of guilt, given the evidence of the dead children, the jury needs to weigh up the relative likelihood of the two competing explanations for the deaths. Which is more likely? Double infant murder by a mother or double SIDS. In fact, double SIDS is much more common than double infant murder.
More generally, it is likely in any large enough population that one or more cases will occur of something which is highly improbable in any particular case. Out of the entire population, there is a very good chance that some random family will suffer a case of double SIDS. This is no ground to suspect murder, unless there was a particular reason why that particular family was likely to harbour a double child killer. It is like convicting someone of cheating the National Lottery simply because they happened to hit the jackpot.
Such miscarriages of justice become even more likely when individually very weak charges are bundled together with other only vaguely related weak cases, to help each other across the line. Active trawling for complainants opens up the additional possibility of related dangers. Our growing understanding of cognitive biases in human perception and decision-making, combined with the deviation of true conditional probability from what common intuition tells us, means that we should be more aware than ever of how easy a miscarriage of justice is. The reality is that the way the law has developed over the last 125 years, and particularly the last 25 years, notably by very severely weakening the required degree of similarity between bundled charges, has made those miscarriages more likely than ever.

The growth of electronic records, including stored texts and messages, also indicates cause for alarm. Recent cases show that exculpatory evidence was contained in the form of texts. In one case a single isolated text provided the vital piece of evidence to prove a false allegation. Which raises the question. How often are false allegations admitted to electronically? Very rarely, almost certainly. So in bringing to light those that we do know of surely indicates that there are many multiples of these false allegations that are not recorded and stored electronically. In a context where police are directed to automatically believe complainants, this is particularly problematic,

Add in the systematic failures of the police and prosecution to reveal exculpatory evidence to the defence. Add in the well-known failures of eyewitness identification evidence. Add in false memory syndrome. Add in cross-contamination of DNA evidence. Add in forced or false confession evidence. Add in arbitrary target conviction rates. Add in confirmation bias by police and the prosecution service. And we have a problem. A very big problem.

As for Sally Clark, she never recovered from the trauma of losing her children and spending years in prison falsely convicted of killing them. She died on 16th March, 2007, of acute alcohol intoxication.

Similar Fact Evidence

http://www.richardwebster.net/similarfactevidence.html

Bundling of charges

https://www.thetimes.co.uk/article/these-bundles-of-charges-pose-a-real-danger-cgzhph7nsf5

Bundling of allegations

https://www.thetimes.co.uk/article/even-rolf-harris-deserved-better-than-this-v3xcvqfxx

Confirmation bias

https://www.thetimes.co.uk/article/this-disgraceful-chief-constable-must-quit-lh5lrfwts

https://www.thetimes.co.uk/article/met-chiefs-perverted-our-system-of-justice-vwgnbl9hb

High-profile miscarriages of justice

http://www.telegraph.co.uk/news/uknews/law-and-order/11075284/When-innocent-men-go-to-jail-miscarriages-of-justice-in-Britain.html

http://news.bbc.co.uk/hi/english/static/in_depth/uk/2001/life_of_crime/miscarriages.stm

https://en.wikipedia.org/wiki/List_of_miscarriage_of_justice_cases

DNA cross-contamination

http://www.telegraph.co.uk/news/2017/12/20/police-doctor-warns-miscarriages-justice-rape-cases-row-dna/

False memory syndrome

https://www.theguardian.com/science/2010/nov/24/false-memories-abuse-convict-innocent

False confession evidence

https://www.theguardian.com/uk/2011/oct/09/false-confessions-sean-hodgson-courts

False eyewitness and identification evidence.

https://www.theguardian.com/uk/2009/aug/18/eyewitness-evidence-wrongful-conviction

Memory, trawling and the misinformation effect.

Memory, Trawling and the Misinformation Effect

Non-disclosure of evidence.

http://www.bbc.co.uk/news/uk-42431171

Click to access Liam-Allen-So-close-to-a-miscarriage-of-justice-Graham-Gilbert.pdf