A proof is a proof. What kind of a proof? It’s a proof. A proof is a proof, and when you have a good proof, it’s because it’s proven.

Jean Chretien

“Mathematicians are like Frenchmen; if you talk to them, they translate it into their own language, and then it is immediately something quite different.”

Johann Wolfgang von Goethe

I particularly recommend this article by Polymathematics about John Conway’s soldiers in the desert problem. I’d never actually seen a proof of this before.

Broadly speaking, Meyer and Finney argue that the statistics tell the following two stories. First, in very serious accidents, airbags save many lives. However, when airbags go off accidentally or incorrectly – not in serious accidents – the act of the airbag exploding can be very dangerous, even fatal. They argue that government bodies have paid too much attention to the first statistic, to the detriment of the latter, even though this may happen more often. To quote:

Here is an analogy to help understand this: If you look at people who have cancer, radiation treatment will improve their probability of survival. However, radiation treatment is dangerous and can actually cause cancer. Making everyone in the country have airbags and measuring effectiveness only in the fatality group, is like making everyone have radiation treatment and looking only at the cancer group to check efficacy. Within the cancer group, radiation will be found to be effective, but there will be more deaths on the whole.

It’s well worth a read, and includes some discussion from a dissenter at the end.

Who want airbags? (PDF)
Mary C MEYER and Tremika FINNEY
Chance, 18:2, 3-16, 2005

Her conviction, and subsequent acquittal, became very famous in the UK, particular because of some of the statistics offered by an expert witness, Sir Roy Meadow. Meadow’s evidence was, basically, that the chance of both of Sally Clark’s sons dying of natural causes was so slim, that one had to assume that they were murdered. The structure of his argument was as follows:

The chance of a randomly chosen child dying or Sids (Sudden Infant Death Syndome, popularly known as cot death) is about 1 in 3000.

Amongst non-smoking, older parents, with at least one wage, this rises to about 1 in 8500.

The chance of both Sally Clark’s sons dying of Sids is about 1 in 8500×8500, or 1 in 73 million.

The chance that the two boys were not murdered is 1 in 73 million.

Meadow’s error has made three errors here: First, he assumed that having one child die of Sids has no affect on the likeliness of a second child dying the same way, hence the 8500×8500 figure. Evidence suggests that, in fact, if one child dies of Sids, the chance of it’s sibling dying that way also is reduced to 1 in 200. Quite a difference from 8500.

Second, the likeliness of innocence given the evidence has been confused with the likeliness of the evidence given innocence. No-one asked what the chance a randomly chosen mother killed her two children. This is known as the Prosecutor’s Fallacy. This is explained superbly by Helen Joyce in an article for Plus magazine. (It’s worth noting that this article was written, basically for schoolkids, whilst Sally Clark was still in prison.)

The third, slightly more subtle, is that whilst Meadow used factors decreasing the likelihood of a Sids death (non-smoking, wage, age), he did not appear to look for factors that might have increased such a likeliness.  In other words, he chose the statistics that best supported his case, and ignored those that didn’t.

That such mistakes that were allowed to go virtually unchallenged in court is almost unbelievable – this is basic, basic statistics. It’s on the GCSE syllabus. Fifteen year old kids should know better than to accept such an argument. That Meadow didn’t know this, the judge didn’t know this and no-one on Sally Clark’s defence team knew this shames them all.

As the Royal Statistical Society stated in a press release (pdf),

[T]here is a real possibility that without proper guidance, and well-informed presentation, frequency estimates presented in court could be misinterpreted by the jury in ways that are very prejudicial to defendants. Society does not tolerate doctors making serious clinical errors because it is widely understood that such errors could mean the difference between life and death. The case of R v. Sally Clark is one example of a medical expert witness making a serious statistical error, one which may have had profound effect on the outcome of the case.

Richard Webster, (in a very fine and detailed article on this subject, well worth a read, see below) is more forceful:

When [Meadow] told the jury that the odds against two of Sally Clark’s children dying of unidentified natural causes (or Sudden Infant Death Syndrome) were 73 million to one, he was leaving the area of his expertise – paediatric medicine – and entering a field in whose basic principles he was completely unschooled – statistical probability. He did so without alerting the jury or the court to the fact he was no longer addressing them as an expert and thus invested a statistical fallacy with all the authority his medical status endued him with.

Meadow was struck off the medical register in July 2005. He was reinstated 7 months later on appeal. Sally Clark spent 20 months in prison, until her conviction was quashed on her second appeal.

Sally Clark’s third son was born while she was on bail. (The jury were not allowed knowledge of his existence during the trial, instead being told the Sally Clark was a hopeless alcoholic who murdered her children because they had ruined her trim figure and career prospects.) He spent much of his early life without his mother, who was wrongly imprisoned for a crime she did not commit, and has now lost her permanently.

References
Richard Webster Roy Meadow, the statistics of cot deaths and other fragments
Ray Hill Cot death or murder – weighing the possibilities (Word document)

Aliens invade the earth and threaten to obliterate it in a year’s time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world’s best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack.

Paul Erdős