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Prosecutor's fallacy
The prosecutor's fallacy is mixing up two questions: how likely is this evidence if the person is innocent, and how likely is it that the person is innocent given this evidence.
Summary
- The two questions can have very different answers. The chance that a dog has four legs is not the same as the chance that an animal with four legs is a dog.
- A test result works the same way. Take a disease that 1 person in 1,000 has, and a test that picks up 9 in 10 cases and is wrongly positive for 1 healthy person in 100.
- Of a million people, 1,000 have the disease and the test finds 900 of them. It also flags 9,990 of the 999,000 who are healthy.
- A positive result therefore means about a 1 in 12 chance of having the disease. The tempting answer is 99 in 100.
- The answer depends on how common the disease was to begin with, called the prior probability, as well as on how good the test is. Bayes' rule is the formula that combines the two.
- In court the fallacy runs like this: these events would be very unlikely if the accused were innocent, so the accused is very unlikely to be innocent.
- The fair comparison is how likely the evidence is under each explanation, guilt and innocence.
- Spotting the fallacy weakens one argument for guilt and says nothing about the rest of the evidence.
In practice
- At Sally Clark's trial in 1999 an expert told the jury the chance of two cot deaths, the sudden unexplained deaths of babies, in a family like hers was 1 in 73 million.
- The Royal Statistical Society, the professional body for statisticians, said the figure had "no statistical basis".
- It added that reading the figure as the chance the deaths were accidental was the prosecutor's fallacy.
- The Court of Appeal overturned her convictions in January 2003, mainly because test results had not been disclosed at her trial. It said the figure should never have gone before the jury in the way it did.
- Lucia de Berk, a Dutch nurse, was convicted in 2003 after a court heard that the chance of so many deaths on her shifts was 1 in 342 million. Statisticians later said it may have been as high as 1 in 25, and she was acquitted at a retrial in 2010.
- In Lucy Letby the prosecution said she alone was present on the unit at every deterioration and death in the case.
- Kit Yates, a senior lecturer in mathematical biology at the University of Bath, argued in 2025 that reasoning from how unlikely so many deaths would be by chance is the prosecutor's fallacy.
- Her convictions stand, and no court has tested that argument. The Criminal Cases Review Commission, which investigates possible miscarriages of justice, is reviewing the case.
Interview questions
- What is the prosecutor's fallacy?
- A patient tests positive on a screening test that is right 99 times in 100. What would you want to know before telling them how likely they are to have the disease?
- A nurse was on duty for every unexplained death on a ward. What would you want to know before drawing a conclusion?
Sources
- rss.org.ukrss.org.uk/…/Report_Healthcare_serial_killer_or_coincidence_statistical_i
- rss.org.ukrss.org.uk/…/Sally-Clark-RSS-statement-2001.pdf
- The National Archives, case lawcaselaw.nationalarchives.gov.uk/…/1020
- The National Archives, case lawcaselaw.nationalarchives.gov.uk/…/748
- theconversation.comtheconversation.com/…/experts-have-challenged-the-medical-case-against-lucy-letby-
- Criminal Cases Review Commissionccrc.gov.uk/…/chairs-statement-on-lucy-letby-application-review
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