P(A | B) = P(B | A) P(A) / P(B)
Let us call event A ‘woman murdered by her batterer’, represented by M,B.
And call event B ‘woman murdered’, represented by M.
A bit further below we will introduce another event ‘a woman battered by her significant other’, represented by B.
William Skorupski and Howard Wainer, in their paper ‘The Bayesian flip: Correcting the prosecutor’s fallacy’ (Royal Statistical Society, August 2015) use the following data, which they in turn got from the Clark County Prosecuting Attorney webpage and a 2010 New York Times article by Steven Strogatz:
•In 1992, the population of women in the USA was approximately 125 million.
•In that year 4,956 women were murdered.
•Approximately 3.5 million of women are battered every year.
•In that same year 1,432 women were murdered by their previous batterers.
The probability of a woman being murdered overall:
P(M)
= 4,956 / 125 million
= 0.00003965
The probability of a woman being battered:
P(B)
= 3.5 million / 125 million
= 0.028
The probability of a woman being murdered by her batterer in the whole population:
P(M, B)
= 1,432 / 125 million
= 0.00001146, or one in 87,290.
The conditional probability of a still living woman subsequently being murdered by her batterer assuming she was indeed already being battered:
P(M, B | B)
= 1,432 / 3.5 million
= 0.0004091, or one in 2,444.
This is the (approximate) ratio that O.J. Simpson’s defence used in his 1994 trial to counter the prosecution’s evidence that Simpson beat his ex-wife and their deduction that this significantly increased his chance of having been her murderer. The defence said that since only one out of 2,500 battered women go on to be murdered, the prosecution’s evidence was of no relevance to the trial, and so the jury should ignore it.
But we will see that this probability P(M, B | B) is not relevant, in this case the ‘defence attorney’s fallacy’. As already stated, it is merely the probability of a battered and still living woman being subsequently murdered before we find any dead body.
What we need to determine is P(M, B | M), the probability that the woman was murdered by her spouse, given that she has indeed been found murdered:
P(M, B | M) = P(M | M, B) x P(M, B) / P(M)
P(M | M, B) is the probability of the woman being murdered, once she has been murdered by her batterer = 1 as already stated.
So:
P(M, B | M)
= 1 x 0.00001146 / 0.00003965
= 0.2890, or a one in 3.5 chance,
of a battered and murdered woman, having been murdered by her batterer.
This would have been an important addition to the trial in light of a wealth of other evidence accusing Simpson of the crime.
However, thanks to the defence attorney’s fallacy amongst other arguments, Simpson was acquitted.
Appendix 5
The ‘ecological factor’
The question ‘would you cross a river that is four feet deep on average?’ comes from Nassim Taleb’s The Black Swan.
As pointed out both in the main story and in the Postscript, the ‘ecological fallacy’ is where the Professor assumed the cot death probability within any single family was the same as the aggregate (or average) probability for all cot deaths, without taking into account conditions specific to individual families (such as the notion of a S.I.D.S. gene).
Even without accounting for a hypothetical cot-death gene, Professor A’s real-life data showed a lot of variability. The data was extracted from the C.E.S.D.I. study (Confidential Enquiry into Sudden Death in Infancy). This analysed 400 sudden infant deaths in the UK over a period of three years from 1993 to 1995 inclusive. The report identified three main factors associated with an increased risk of death in a particular household, which were (1) the presence of smokers, (2) younger mothers under twenty-seven, and (3) whether the household had no wage earner.
We see an incredible variation depending on which factors are present, ranging from one cot death in 8,543 live births, to one in 214 births:
S.I.D.S rates for different factors based on the data from the C.E.S.D.I. S.U.D.I. Study:
S.I.D.S. incidence in this group:
Overall rate in the study population
One cot death for each 1,303 live births
Rate for groups with different factors:
Anybody smokes in the household
Nobody smokes in the household
One in 737
One in 5,041
No waged income in the household
At least one waged income in the household
One in 486
One in 2,088
Mother less than 27 years old
Mother 27 years old or more
One in 567
One in 1,882
None of these factors
One of these factors
Two of these factors
All three of these factors
One in 8,543
One in 1,616
One in 596
One in 214
With this variation in data, it was statistically very dangerous to assume that the one in 8,543 average for the Richardson’s type of household applied exactly to the Richardsons.
This is also a fair proof that different environmental factors (amongst others) can influence the incidence of S.I.D.S.:
•Hence cot deaths are not random, even if they appear to be.
•Hence you cannot assume statistical independence by squaring the individual probability to get the probability of two events happening.
•There is statistical dependence: the arrival of one cot death suggests higher odds of a second cot death.
The incidence of a possible cot death gene (not identified in this study) could only accentuate this statistical dependence.
It is interesting to observe the real-life contradictions:
•While the eminent medical Professor A testified in several trials that there was no evidence of increased risk of having a second cot death once a first had happened.
•We found, meanwhile, that the much more intuitive and down-to-earth NHS healthcare workers always imposed strict controls on second babies born into families having previously suffered a cot death, in particular the use of a sleep apnoea alarm and giving CPR lessons to the parents…
No prizes for guessing who got it right.
About the Author
Michael Carter grew up in Norwich, then studied engineering at Cambridge and Ocean Engineering at UCL, before designing and installing offshore oil platforms - just like the story’s narrator.
Later he did his own Brexit and moved to France, and after an MBA at INSEAD (top of his class) he spent the rest of his career in senior management positions in electrical multinationals (CEO of Socomec Group and Vice President at Legrand), before retiring in 2018 to go sailing, mountaineering and write thought
provoking books.
He lives in Grasse in the South of France, is married, and has three daughters, two dogs and a sailing boat.
Although this is his first novel as a writer, he has honed his writing skills over the years, since managing multinationals takes a lot of bi-lingual skills in communicating new concepts in as interesting a manner as possible.
Notes
1See the appendix for the derivation of Bayes’ theorem from first principals, and then the ‘Bayesian flip’ to get this equation.
The Mathematical Murder of Innocence Page 13