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  \textbf{\Large What is the chance of your being guilty?}
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  \textbf{JOHN KAY}
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The chance that a random sample of DNA would match that of  O.J.\
Simpson was put at one in 4m. Long odds: but, as Johnnie Cochran, Mr 
Simpson's counsel, explained to the jury, there are 20m people in the
Los  Angeles area. Mr Simpson was therefore one of several people whose
blood  might be matched to the scene and he could not be guilty beyond
reasonable  doubt.  

The probability that one of Sally Clark's children would  suffer a cot
death was estimated at one in 8,500. The chance of two  unrelated
incidents in a single family is one in 8,500 multiplied by  8,500, or
one in 73m---far larger than the number of new mothers in the  UK. The
defence account of events was thus so improbable as to leave  little
doubt of her guilt.  

Mr Simpson was acquitted and Ms Clark  convicted. But the plausible
arguments described above are fundamentally  flawed. Ms Clark, who was
sentenced to life imprisonment in 1999, was  freed by the UK's court of
appeal in January. The error in her case is now  sufficiently familiar
to lawyers to have acquired the title of ``the  prosecutor's fallacy''. 


Thomas Bayes, an 18th-century Nonconformist  clergyman, discovered that
his game of billiards was improved by an  understanding of contingent
probab\-ilities---the likelihood that an event  will occur if some other
event has already occurred. He went on to derive  Bayes' theorem, by
which contingent probabilities are calculated.  Contingent probabilities
are central to understanding the statistics of  the Simpson and Clark
cases.  

The prosecutor's fallacy is the  assertion that, because the story
before the court is highly improbable,  the defendant's innocence is
equally improbable. But all accounts of  events in high-profile legal
cases are highly improbable. That is why they  are high-profile legal
cases. The courts do not hear reports of happy  families and normal
behaviour. Their services are required only for  bizarre and unlikely
incidents---such as the saga of Nicole and O.J.\  Simpson and the
tragedy of the Clark family.  

So juries are not  asked to decide whether the events before them are
out of the run of  everyday experience. They are asked to decide on the
most probable  explanation of improbable events. This is how the
analysis of contingent  probabilities developed by Bayes comes into
play.  

Bayes' theorem  resolves the Simpson case quite easily. If Nicole
Simpson was not murdered  by OJ, she was murdered by someone who,
although not her former husband,  had DNA matching that of her former
husband. This is so unlikely as not to  constitute a basis for
reasonable doubt.  

And Bayes' theorem shows  that probability and statistics cannot decide
in cases like Mrs Clark's.  Unexplained infant death is rare but when it
occurs it more often results  from natural causes than from murder. A
single incident should therefore,  in the absence of other evidence, be
treated as natural. But what of a  series?  The simple theory of
probability, as applied by Johnnie Cochran  and Professor Sir Roy
Meadow, an expert witness in the Clark case,  multiplies the
probabilities of a series of events together on the  assumption that
they are unrelated. But it is unlikely that these effects  are
unrelated.  Murderous intent may persist in the mind of evildoers. 
Genetic and environmental influences that cause death may affect
siblings.  We simply do not know the relative significance of these
factors, so we  must look to other evidence - such as the blood sample
that, after Ms  Clark had spent two years in prison, indicated that her
son Harry had  probably died of a respiratory infection.  

The lesson is that the  simple rules of logic and arithmetic involved in
probability theory are  hard to handle. I have never talked to an
audience about a range of  possible scenarios without being asked: "But
what do you really think is  going to happen?" The human mind,
uncomfortable with uncertainties and  probabilities, prefers to deal
with narratives and stereotypes. That is  true of trial juries and
business people, investors and politicians. And  it is one reason they
keep making mistakes that damage their reputations,  their wealth and,
sometimes, the lives and freedom of other people. 

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\textit{Financial Times}, 20 June 2003, p.\,21;
\texttt{http://www.johnkay.com/society/287}.

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