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### The Poisson probability distribution sheds more light on the Lucy Letby case

I am not a lawyer. Nor am I a medical expert. But I have what I think is a decent grounding in statistical analysis which permits me a useful perspective in relation to so-called 'cluster cases', legal cases (often pertaining to the medical world) where there is an unusually high number of events (e.g. deaths of patients) that is deemed suspicious (the Lucy Letby case in which she is accused of murdering 7 neonatal babies on the HDU (high dependency unit) at the Countess of Chester Hospital and attempting to murder a further 10 on 15 occasions is such a case). The probability distribution known as the Poisson distribution is particularly useful in such cases and in Letby's case yields some remarkable findings.

What is the Poisson distribution?

The Poisson distribution describes the distribution of rare events (in time or space) where events are unrelated to each other and the likelihood of an event occurring (in a unit of time or space) is fixed.

Real world examples of events that are 'approximately Poisson distributed' ('perfect' does not exist in the real world) are the number of calls received by a call centre in one minute, the number of stars that can be seen through a telescope, the number of zircon crystals in a cubic inch of sand, the number of radioactive decays per second (note that stars and zircon crystals are 'events' in space while telephone calls and radioactive decays are events in time).

If, based on historical average, the likely number of calls received by a call centre in one minute is 1.2 (among the millions of telephones from which calls could be received i.e. rare) then the probabilities of receiving 0, 1, 2, 3, etc in one minute can be calculated using a mathematical formula (see https://en.wikipedia.org/wiki/Poisson_distribution), and are as below:

Number of calls rec'd Probability (%)

in one minute

0 30.1%

1 36.1%

2 21.7%

3 8.7%

4 2.6%

>5 0.8%

Exactly the same probabilities would pertain in the case of zircon crystals where the average number in a cubic inch of sand was 1.2 (among the hundreds of thousands of grains of sand i.e. rare). Or 1.2 radioactive decays (among billions and billions of atoms) or 1.2 stars in the viewfinder of a telescope (that could contain many more). Cool, huh?

In relation to neonatal deaths (which thankfully are rare) you might expect the number per year (at a hospital, in a particular region, or the country) to decline over time given improvements in care, but the improvement will not be rapid (the rate for England as a whole improved from 1.28 per year per 1,000 live births in 2013 to 0.97 in 2020, a rate of 4% per annum - see MBRRACE UK Perinatal Surveillance Reports 2013 to 2020 on https://www.npeu.ox.ac.uk). So the likely number per year can be considered fixed. And neonatal deaths are going to be independent of each other (the condition of a particular baby is unrelated to the condition of another).

So, neonatal deaths are going to be approximately Poisson distributed. Unless, that is, there is something else going on that is influencing things e.g. the presence of a serial killer, a staff shortage, poor medical care, or contamination of some sort.

Lucy Letby was charged with 7 murders during a 13 month period: 3 in June 2015, 1 in August 2015, 1 in October 2015, and 2 in June 2016 though, strangely, there was no record of the deaths in August and October 2015 in the hospital's official records (see https://www.whatdotheyknow.com/request/521287/response/1255362/attach/3/FOI%204568.docx?cookie_passthrough=1).

The hospital's neonatal deaths data can be combined with the information pertaining to the murder charges in the chart below.

If we remove the 7 alleged murders from the above, we end up with data pertaining to non-suspicious deaths, as below. This series should be Poisson distributed (unless in addition to a serial killer there was something else going on e.g. a staff shortage, poor medical care, or contamination of some sort.)

Now, the period we are interested in is the one during which Letby is alleged to have been on her killing spree, from June 2015 to June 2016. There were 9 non-suspicious deaths during this period, from July 2015 to March 2016. We want to know the probability of there being nine deaths during the nine month period in question (July 2015 to March 2016).

To determine this we calculate the expected number per month, which we can calculate based on the average outside of the nine month period in question (the answer is 0.18 per month). To calculate the probability of nine deaths in a nine month period, we first multiply this monthly average by nine to get the expected number of deaths in any nine month period (0.18 times 9 equals 1.65). We then use this average to calculate the probability of there being nine deaths during the nine months from July 2015 to March 2016.

The answer?

0.0048% or 1 in 20,845!

In other words, there is a 1 in 20,845 (0.0048%) probability that the nine deaths occurred by chance. Conversely, there is a 20,844 in 20,845 probability (99.9952%) that the nine did not occur by chance i.e. that in addition to a serial killer there was some other factor at work.

Given that it is extremely unlikely that there were two unrelated factors simultaneously causing the number of deaths in 2015 and 2016 to be elevated (serial killer plus something else) and that the probability of the 'non-serial killer factor' is virtually 1, this must surely cast doubt on the prosecution's assertion that Letby is a murderer (this is neither a legal nor a medical argument but a purely statistical one).

There is another possibility, namely that Letby was responsible for all or many of the nine 'non suspicious' deaths but left no trace. However, given that it is clear the prosecution's case was driven first by roster data (note similarities with Lucia de Berk case) and subsequently by alleging various methods of killing (air embolism, insulin administration, other acts of harm, etc) it would seem that the reason that Letby was not charged with any of them was that she wasn't on duty (surely if Letby had been on duty, and given the inconsistency in the medical evidence presented by the prosecution for the 22 charges and the circumstantial nature of it, a way would have been found to charge her with them. Perhaps the reason parents did not demand in the press they be investigated is that they were told by the police Letby was not on duty so could not be connected with them). Moreover, if Letby was clever enough to use undetectable methods in relation to some or all of the nine deaths, why also use highly detectable and risky methods such as insulin administration in others? One would also have to question, if Letby was so adept, why she failed on 15 occasions (the attempted murder allegations).

The only way the case makes sense statistically is if Letby is not a murderer.

The views expressed in this communication are those of Peter Elston at the time of writing and are subject to change without notice. They do not constitute investment advice and whilst all reasonable efforts have been used to ensure the accuracy of the information contained in this communication, the reliability, completeness or accuracy of the content cannot be guaranteed. This communication provides information for professional use only and should not be relied upon by retail investors as the sole basis for investment.

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