I recently attended a lecture given by Professor Raymond Hill on Mathematics and the Law. It focused on a number of cases where a misunderstanding of probability and statistics in particular had led jurors to acquit or convict in the teeth of the evidence presented, to prosecutors to construct cases which made no logical sense, to expert witnesses to mislead and for judges to misdirect juries.

One particular case he mentioned concerned the tragic death of two babies born to Sally Clark, a solicitor from Cheshire, within 2 years of each other. Sally was charged with the murder of both babies once the second had died. At her trial in November 1999, Professor Meadow, a paediatrician but clearly not a mathematician, claimed that, in this case, the chance of two babies dying from sudden infant death syndrome or cot death was 1 in 73 million. This figure came from a study of the deaths of all babies in five regions of England between 1993 and 1996, which estimated that the chance of a randomly chosen baby dying a cot death fell, if the child was from an affluent non-smoking family with the mother aged over 26 like Sally Clarke’s, from 1 in 1303 to 1 in 8543. Piling travesty upon travesty, the chance of Sally Clark suffering two cot deaths was then calculated as 1 in 8543 times 1 in 8543, which is where the 73 million figure comes from. Sally Clarke was convicted on the basis of this ludicrous kangaroo statistical “evidence” and spent over 3 years in jail and needed two appeals before she was finally cleared. A full account of the case, and how Professor Hill went about presenting the absurdity of it, can be found here.

As Blaise Pascal wrote: “You always admire what you really don’t understand.”

Mathematics and law can come into conflict for a number of reasons, but one thing that doesn’t help is that they share a lot of the same words. Proof, for instance. But where this means an immutable truth in mathematics, as true today as it was thousands of years ago and as it will be thousands hence, proof in law will depend on the time in which the trial takes place and the burden of proof required. When there was the threat that Syria would be bombed by the UK and US, some opponents used the idea that you shouldn’t pass a death sentence on whoever would be standing under the bombs unless the Syrian regime had used chemical weapons “beyond reasonable doubt”. I saw one estimate of this as an 80% probability, however I have since seen 99% probability presented as a definition. So proof in law is a more elastic concept.

As a pensions actuary, I have had my own, rather different, problems with the interaction of mathematics and law. Defined benefit pension schemes are mathematical constructs as well as legal constructs. If you do A and B and earn C, then the pension scheme to which you belong should deliver benefits to you of D. However a pensions lawyer would see it rather differently, in terms of obligations of certain parties towards other parties under the legal construct of a trust.

When drafting pension scheme rules, lawyers often have to set up quite complex conditional relationships between possible events and outcomes. It is quite possible for some of these to be left out (in which case we hear that “the trust deed and rules are silent”), and also for them to be included but in a way which displays a certain amount of ignorance of mathematical logic, meaning either that the rules are very difficult to implement or have unintended consequences. This generally then creates work for a different set of lawyers down the track.

As a result, actuaries have long accepted that trying to interpret the rules of any pension scheme without legal advice is just asking for trouble. And the list of legal disclaimers actuaries populate their reports with grows year on year as a new threat of future second guessing emerges. There is therefore certainly considerable respect for the importance of the legal elements of the construct of a pension scheme by actuaries, if not always full understanding. Unfortunately, the same does not always hold in reverse. I have seen numerous examples of rules drafted without the mathematical elements of the construct fully taken into account by the drafters:

- benefits either too ambiguous to value or in contradiction with each other;
- double revaluation of benefits built into the rules in one instance;
- elements of scheme design which would obviously need to be reviewed in the future, like commutation factors of 9 to 1 for instance, hard coded into rules so that they can only be changed by a deed of amendment.

Actuarial input into any issue around a pension scheme is frequently dismissed by lawyers as “crunching the numbers”. I think most of them would be mortally offended if an actuary turned to them and asked them to crunch the words.

Pensions lawyers and actuaries need each other if pension schemes are going to work properly. And they need to understand each other rather better too.