Case Study 2: *R v. Sally Clark* — When the Prosecutor's Fallacy Convicts the Innocent

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Case Study 2: R v. Sally Clark — When the Prosecutor's Fallacy Convicts the Innocent

A structured analysis of a documented wrongful conviction, from a complementary angle: a case where the statistical error this chapter names was committed by an expert witness in court, contributed to convicting an innocent person, and was publicly repudiated by a national statistical body. The facts below are drawn from the public appellate record and the Royal Statistical Society's published statement, both Tier-1 sources. Handle the subject — the deaths of infants — with the sobriety it demands.

Why this case teaches the chapter's limits

People v. Collins (Case Study 1) showed a court catching the prosecutor's fallacy. Sally Clark shows what happens when it is not caught in time. It is not a DNA case — the contested statistic concerned the probability of two infant deaths in one family — but it is the clearest documented illustration of the exact error in §9.4 doing exactly the harm this book exists to prevent: an authoritative-sounding number, a transposed conditional, and an innocent person imprisoned. The mechanism transfers directly to DNA, which is why it belongs here. If you understand how the statistic failed Sally Clark, you understand why the honest script in §9.6 is not pedantry but protection.

Background

Sally Clark was a solicitor in England whose two infant sons died, the first in 1996 and the second in 1998, each found unresponsive at home some weeks after birth. Both deaths were initially treated, by at least some of those involved, as possible cases of sudden infant death syndrome (SIDS) — unexplained natural deaths. After the second death, Clark was arrested and charged with murdering both children. She was convicted in 1999 and imprisoned. She had not harmed her children.

The statistical evidence and the error

At trial, a pediatrician testified as an expert. To convey how unlikely it was that both infants had died of natural causes, he offered a striking figure: the probability of two SIDS deaths occurring in a family like the Clarks' was, he said, about 1 in 73 million. The number was arrived at by taking an estimated probability of a single SIDS death in such a family (illustratively, around 1 in 8,500) and squaring it — multiplying it by itself — to get the probability of two.

This single figure carried two compounding errors, both of which this chapter has taught you to spot.

Error 1 — An invalid multiplication (the independence assumption, again)

Squaring the single-death probability assumes the two deaths were independent events — that the second death was no more likely given the first. But SIDS is not randomly distributed: shared genetic factors, shared environment, and shared family circumstances mean that a family who has suffered one such death may be at elevated risk of another. The deaths are plausibly correlated, and multiplying correlated probabilities — exactly the §9.1 caution, and exactly the Collins error — produces a figure far smaller, and far more damning, than the evidence supports.

⚠️ Junk-Science Alert The "1 in 73 million" figure is Collins' product-rule error reborn in a medical courtroom. Independence is the hidden load-bearing assumption behind every multiplied probability. For carefully chosen DNA loci, independence is established by population-genetics research. For two infant deaths in one family, it was simply assumed — and the assumption was almost certainly false. The lesson recurs: whenever you see a tiny probability built by multiplication, interrogate independence before you believe the number.

Error 2 — The prosecutor's fallacy

The second error is the one that names this chapter, and it is the more dangerous because the "1 in 73 million" figure was heard by the jury as the probability of Sally Clark's innocence. The figure — even if the multiplication had been valid — was at most a statement about how rare two natural infant deaths are. It is $P(\text{two deaths} \mid \text{natural causes})$. It is not $P(\text{innocent} \mid \text{two deaths})$. To reach a probability of innocence, one would have to weigh the rare event "two natural deaths" against the also rare event "a mother murders both her infants," together with everything else known about the family — precisely the Bayesian comparison of §9.5. Presenting "1 in 73 million" as the chance she was innocent transposed the conditional and skipped the comparison entirely.

⚖️ In the Courtroom The Royal Statistical Society — the United Kingdom's professional body for statisticians — took the extraordinary step of issuing a public statement expressing concern about the statistical reasoning used in the case. It explained that the "1 in 73 million" figure was both arrived at illegitimately (the independence error) and, more fundamentally, misinterpreted: the probability of the evidence under one hypothesis is not the probability of that hypothesis given the evidence. That a national statistical society felt compelled to intervene in a criminal case tells you how grave, and how recognizable, the error was. It is the prosecutor's fallacy of §9.4, identified by the very experts whose field it misrepresents.

There was a further, separate failure in the case unrelated to statistics — undisclosed medical evidence bearing on the cause of the second child's death — and that non-disclosure featured heavily in the eventual appeal. But the statistical testimony is what makes the case immortal in this field: it is the textbook example of how a number, wrongly built and wrongly interpreted, can overwhelm a jury's judgment.

Outcome

Sally Clark's first appeal failed; a second succeeded. In 2003 her conviction was quashed and she was released, after more than three years in prison for crimes that never occurred. The appellate court's reasoning rested substantially on the undisclosed medical evidence, but the misuse of statistics was a central part of the case's notoriety and a catalyst for reform in how probabilistic evidence is presented in court. Sally Clark never recovered from the ordeal; she died a few years after her release. The case stands as one of the most cited modern examples of statistical evidence causing a miscarriage of justice.

The lessons for forensic DNA

The contested statistic was not a DNA match probability, but every lesson transfers without alteration:

A multiplied probability assumes independence — verify it. The squaring of the SIDS probability, like the product rule in Collins, was invalid because the events were not independent. The same caution guards the legitimate multiplication behind a DNA profile frequency.
The probability of the evidence is not the probability of the hypothesis. "1 in 73 million for two natural deaths" is not "1 in 73 million that she is innocent" — exactly as "1 in a billion that a random person matches" is not "1 in a billion that the defendant is innocent." The transposed conditional is the same error in both rooms.
A rare event must be weighed against the alternative, which may also be rare. §9.5's Bayesian structure is the antidote: the rarity of innocent two-death cases means nothing until it is weighed against the rarity of a double infanticide and the rest of the evidence. A number presented without its comparison is an argument with half its steps hidden.
Even a sincere expert can commit the fallacy. The witness was not a fraud; he was a physician reaching outside his statistical competence. The error is seductive precisely because intelligent, honest people commit it. This is why §9.6 reduces the safeguard to a memorizable rule — report the strength of the evidence, never the probability of guilt or innocence — and why a forensic scientist must police their own language even when no one is trying to deceive.

For your own practice, Sally Clark is the case to remember at the moment of testimony. When you are tempted, or invited, to translate a match probability into a probability of guilt, recall that a national statistical society once had to publicly correct exactly that translation — after it had already helped imprison an innocent woman. The rule that would have prevented it is the rule you will follow.

Discussion questions

The "1 in 73 million" figure contained two distinct errors. Name each, and explain which one would still apply if the single-death probability had been perfectly accurate.
Translate the prosecutor's fallacy in this case into the language of conditional probabilities. What was $P(\text{evidence} \mid \text{hypothesis})$ here, and what was the jury led to believe it represented?
Using the Bayesian framework of §9.5, explain what comparison the "1 in 73 million" figure left out. Against what alternative hypothesis should the rarity of two natural infant deaths have been weighed?
The expert in this case was a physician, not a statistician, testifying outside his area of statistical competence. Connect this to the chapter's argument (and the rule in §9.6) about why even sincere experts need a disciplined script. What structural safeguard might have prevented the error?
Compare Sally Clark with People v. Collins (Case Study 1). In Collins the court caught the fallacy on appeal and reversed; in Clark it took two appeals and years in prison. What does the difference suggest about how reliably the legal system catches statistical errors — and what does that imply for the expert's responsibility at the moment of testimony?
Apply the lesson to DNA. Suppose an analyst testifies that a mixture LR is 1 billion and adds, "so it is a billion to one that anyone other than the defendant contributed." Identify the error, and write the sentence the analyst should have said instead. Then connect it to the detective's claim in this chapter's Cold Case File.