In all probability
21 June 2004
16 September 2013
22 July 2013
1 August 2013
10 March 2014
20 February 2014
HG Wells said we should get used to statistics, as “statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write”. The quote was repeated in an article referring to the trial of Sally Clark, found guilty in 1999 of the murder of two of her children. The Court of Appeal quashed her conviction last year. The nub of this judicial error was the high degree of “improbability” – one in 73 million – which Sir Roy Meadow attributed to the probability that two cot deaths could take place in the same family.
His calculation was simple – if there is one cot death in every 8,543 babies, the chance of two occurring is the square of that probability. What Meadow did not share with the jury were his tacit assumptions that the cot death of a baby is due only to chance, that all babies have the same probability of suffering a cot death, and finally, that the two deaths were independent of each other. If these assumptions do not hold true then the probability of both cot deaths happening would have been significantly higher, and so would certainly not have weighed so heavily in the jury’s verdict.
Our decisions depend on common sense, context, the possibility of other explanations and additional information provided by the facts themselves. Appealing to “common sense” – by definition that of the majority – brings its own consequences, such as giving more probability than its “due” to the number 15237 in a lottery than to 44444. Common sense alone may not be up to dealing with concepts of probability.
Probability should help us decide, but in context and in conjunction with other evidence and information. ‘Bayes’ Theorem’, which was declared “inappropriate” to use with a jury by the Court of Appeal in 1996, was designed to evaluate a priori probabilities against available information, quantifying the process and thus aiding decision-making in complex situations. To declare “inappropriate” something that any law student can assimilate in half an hour and be required to rely solely on common sense, seems quite obscene.
Even more remarkable, judges and statisticians use identical strategies to solve similar problems. They make decisions regarding
the “null hypothesis” (the presumption of innocence) on the basis of evidence or samples and this leads them to take risks. They run two kinds of risk: one, of getting it wrong by rejecting this hypothesis; and the second, of getting it wrong by not doing so. And because these risks are contradictory, the null hypothesis establishes this presumption and tries to protect it.
A trial does not prove innocence. Rather, it infers it by not rejecting its presumption. On the contrary, rejecting it in this type of reasoning might lead to adopting fallacious arguments if equally plausible alternative explanations could be found. Common sense led the Sally Clark jury to confuse slight probability with impossibility, and not to put that figure in context.
Would the knowledge of probability concepts improve the skills of our solicitors and judges? It could encourage reasoning that would interpret the norm more intuitively and with greater imagination, and would probably help to develop alternative learning styles. When we promote the use of one particular side of the brain, precisely the side that a computer can most easily replace, is this really helping us to learn how to learn?
Joan Manuel Batista-Foguet is a director of research and a professor of quantitative methods at ESADE Business School