*BMJ* 2000;320:2-3 ( 1 January )

*Doctors and lawyers should get probability theory right*

In a recent case of DNA evidence the probability of a chance match was quoted as
20 million to one. The accurate statementthat^{ }the defendant or two other unknown people in
the United Kingdom^{ }could have committed the offenceis much less impressive. Other^{ }evidence
was overwhelming, but this may not always be true, especially^{ }with matches from
DNA databases. Even more problematic than the^{ }issue of presenting statistical
evidence fairly is the problem^{ }of getting it^{ }wrong.

On 9 November at Chester Crown Court Sally Clark, a Cheshire solicitor, was
convicted, **by** 10-2 majority, of
smothering her^{ }two infant children. With conflicting forensic evidence, the
Crown's^{ }case was bolstered **by** an
eminent paediatrician testifying that^{ }the chances of two cot deaths happening
in this family was vanishingly^{ }small1 in 73 million. This seriously misunderstands
probability^{ }theory. It is speculation whether Sally Clark would have been^{ }acquitted
without this evidence. But with this **mathematical**
error^{ }prominent the **conviction** is^{
}unsafe.

Imagine an archery target with two arrows sticking in the very centre of it. This
provides greater evidence of the skill of^{ }the archer if the target was in place
before the arrows were fired^{ }than if it was drawn around them afterwards.
Probability theory^{ }requires calculation of the probability not only of the
event^{ }in question but also of all events that are as extreme or more^{ }extreme.
When the target is drawn first you calculate the chance^{ }of both arrows hitting
the centre of the target. But when the^{ }target is drawn round the arrows
afterwards you calculate the^{ }chance of both arrows hitting the same point,
whatever that point.^{ }With two independent arrows one probability is the square
of the^{ }other.

Suspicion was drawn to Sally Clark **by** the
occurrence of two deaths so the probabilities should not have been squared. The^{ }odds
of 1 in 73 million shrink to 1 in 8500. But this figure is^{ }itself
meaningless. There is in fact a wall full of arrows with^{ }the target drawn
around the two that are close together and the^{ }others ignored. **Mathematical** formulas for this situation often^{
}surprise people. For example, with only 23 people in a room the^{ }odds
are better than 50% that two of them have the same^{ }birthday.

From whole population data Reese calculates the square of the population risk of cot
death as 1 in 2.75 million.^{1} There^{ }are
378 000 second or subsequent births each year in England.^{ }So if cot deaths
are random events two cot deaths will occur in^{ }the same family somewhere in
England once every seven years. But^{ }cot deaths are not random events. There
have been several studies^{ }of recurrence. At least one study did show no
increase in recurrence^{ }rates.^{2} But several others
showed recurrence rates about five^{ }times the general rate,^{3-5}
implying recurrence somewhere in England^{ }about once every year and a half. Two
studies showed even higher^{ }rates. ^{6} ^{7}

The fact that studies of recurrence have been done means this event is not vanishingly
rare. In a case series of recurrent^{ }infant death Emery classified two cases as
recurrent cot death^{ }out of 12 cases occurring in Sheffield in
20 years.^{8} Wolkind^{ }et al found five cases in
their unsystematic English case series^{ }of 57 recurrent infant deaths.^{9} Both these studies distinguished^{ }cot death from
accident, illness, murder, and^{ }neglect.

The prosecution used the figure of 1 in 73 million rather than 1 in
2.75 million because of the family's affluence. Yet taking^{ }data from an
epidemiological group and applying it stereotypically^{ }to all members is an
example of the ecological fallacy. Social^{ }class is a complex reality of
interassociated circumstanceseducation,^{
}work, income, lifestyle, culture, contacts, residence, opportunities,^{ }social
class of origin, etcstatistically
summarised for use in^{ }population studies **by**
selecting the one variable which performs^{ }best as an indicator. This does not
mean that individuals have^{ }the attributes of the statistical^{ }group.

Guidelines for using probability theory in criminal cases are urgently needed. The
basic principles are not difficult to understand,^{ }and judges could be trained
to recognise and rule out the kind^{ }of misunderstanding that arose in this case.
Never again must^{ }**mathematical**
error be allowed to conflict with **mathematical**
fact^{ }as if each were a legitimate expert^{ }view.

What is our profession's responsibility for the quality of expert evidence given **by** doctors? Medical evidence is trusted,^{ }and
we must retain that situation and ensure that it is not abused.^{ }It is possible
to be an extremely good doctor without being numerate,^{ }and not every eminent
clinician is best placed to give epidemiological^{ }evidence. Doctors should not
use techniques before they have acquainted^{ }themselves with the principles
underlying^{ }them.

When errors occur we expect them to be admitted, learnt from, and corrected. Should
clinical governance extend to the courtroom?^{ }Expert witnesses can hold a
substantial part of defendants' lives^{ }in their hands. Defendants deserve the
same protection as^{ }patients.

**Stephen J Watkins**

Stockport Health Authority, Stockport SK7 5**BY**

Acknowledgments

Predisposing biases: SJW is a vice president and immediate past president of the
Medical Practitioners' Union, which is predisposed^{ }to support the civil
liberties movement. He has no personal acquaintance^{ }with people involved in
this^{ }case.

1. | Reese A. In Statistics and justice. www.stats.gla.ac.uk/allstat/. Accessed
November 1999. |

2. | Peterson DR, Subotta EE, Dubing JR. Infant mortality among subsequent
siblings of infants who died of sudden infant death syndrome. J Pediatr 1986; 108:
911-914 |

3. | Oyen N, Skjaerven R, Jurgens LM. Population-based recurrence risk of
sudden infant death syndrome compared with other infant and foetal deaths. Am J
Epidemiol 1996; 144: 300-305 |

4. | Guntheroth VG, Lohmann R, Spiers PS. Risk of sudden infant death syndrome
in subsequent siblings. J Pediatr 1990; 116: 520-524 |

5. | Irgens LM, Skjaerven R, Peterson DR. Prospective assessment of recurrence
risk in sudden infant death syndrome siblings. J Pediatr 1984; 104: 349-351 |

6. | Froggart P, Lynas MA, McKenzie G. Epidemiology of sudden unexpected death
in infants ("cot death") in Northern Ireland 1971. Br J Soc Prev Med
1984; 25: 119-134 |

7. | Beal SM, Blundell HK. Recurrence incidence of sudden infant death
syndrome. Arch Dis Child 1988; 63: 924-930 |

8. | Emery JL. Families in which two or more cot deaths have occurred. Lancet
1986; i: 313-315 |

9. | Wolkind S, Taylor EM, Waite AJ, Dalton M, Emery JL. Recurrence of
unexpected infant death. Acta Paediatrica 1993; 82: 873-876 |