The Mathematical Murder of Innocence

by Michael Carter

  “No.”

  “What transmits it?”

  “A bite from a mosquito that has previously bitten a malaria carrier,” replied our medical expert, sounding puzzled by my question.

  “Why is it called ‘malaria’?”

  “I’m not sure I know.”

  “That’s a bit disingenuous, Professor Goodwin,” I scolded kindly. “‘Mal’ means ‘bad’, ‘aria’ means ‘air’. ‘Bad air’. Did they not call the disease malaria because they originally thought that ‘bad air’ was infecting people?”

  “Yes, I think you’re right.”

  “So,” I asked, “they had a completely wrong hypothesis – bad air – simply because they did not know?”

  “Yes.”

  “Did everybody who got bitten by a mosquito catch malaria?”

  “No, of course not. As I said, the mosquito had to have previously bitten a malaria carrier.”

  “So, there were two combined reasons.” I counted them off with my thumb and forefinger. “One, to be bitten by a mosquito; two, that the mosquito had to have previously bitten a carrier?”

  “Yes.”

  “Two variables combined. We know now. But we didn’t know a couple of centuries back, did we? Each variable is necessary but on its own not sufficient. Yes?”

  “Yes,” confirmed Goodwin rather wearily, as if to suggest to everybody that I was going off at a tangent.

  “Well,” I countered, “I argue the same for S.I.D.S. Many variables could contribute. You already know one: smoking. But then, not all smokers’ children die of S.I.D.S., do they?”

  “No.”

  “It increases the risk but is not alone the cause. And indeed, non-smoker families can also be afflicted by S.I.D.S.; less than smokers’ families, but afflicted nonetheless?”

  “Yes.” Again, the affected weariness in his voice.

  “So, I repeat what I said before, just because we do not know the causes does not mean that there are no causes. It simply means we don’t know. Therefore, it’s not random, it’s due to these causes – that we don’t know – that makes it appear random. Do you agree?”

  Goodwin decided to deflect the question. “I don’t know what you are trying to get at. The result is clearly random.”

  “I strongly disagree,” I insisted. “You are mixing up appearing to be random and really being random. If the causes are not random, the same causes would quite likely exist a second time around. They could be environmental, they could be genetic, and there are probably several causes, each one contributing to trigger this unlikely event. But if the first baby dies of cot death, quite probably this unlikely combination of causes is also present for the second baby. That means that the risk of the second baby dying as well is much higher. Thus, there isn’t statistical independence, we have the opposite: statistical dependence.”

  I decided to plough on. “And if there are several causes that must be combined to trigger the event, it is almost impossible to test for them. There are too many, and each one individually doesn’t give us the result. It’s only when they happen together that tragedy strikes. So, since you can’t test for them, you don’t know them, so the result appears random.

  “Let’s take your example of a dice,” I said to Goodwin. “Without anyone knowing, the side opposite the six – the number one – is weighted. Mrs Richardson rolls the dice for the first time. She gets a six. You said her getting a six the first time does not mean she has more chance of getting a six the second time, because each roll of a dice is statistically independent. But you were assuming her dice is a normal random dice that is fair and not weighted.

  “But,” I continued, “if we then told you, ‘by the way, her dice is weighted’, you would immediately conclude, the weighting seems to favour the six, therefore there is more chance of her getting a six the second time. Suddenly, with this information, the rolling of the dice becomes statistically dependent – dependent on the result of the first roll.”

  I noticed the judge was slowly nodding his head in understanding. That was good news, I thought. Maybe he gambles with dice in his spare time.

  Chapter Ten

  “Let’s come back to your data on S.I.D.S.,” I said, looking at Goodwin. “On average you say only one out of 8,500 affluent, non-smoking families with older mothers suffer this tragedy. On average, right?”

  “Yes,” he replied.

  “Professor, you remember what we said earlier: an average is made up of a collection of higher probabilities than one in 8,500 for some families, and lower probabilities for other families.

  “Now without our originally knowing it,” I went on, “the Richardson’s dice is weighted due to environmental or genetic reasons, or quite possibly both. Mrs Richardson rolls the dice. Tragedy strikes, she loses her first baby. Given this information, we can imagine that there is a higher than one in 8,500 chance of tragedy striking a second time. Quite possibly a lot higher.”

  “Sorry, I beg to differ,” argued Goodwin. “First, the police supplied my team with a full check on what they found in the environment in the Richardsons’ home. Our analysis found no reason to suppose that this was contributing to the deaths. Secondly, there is clear evidence of no genetic history of infant deaths in either families. To use your very own logic of a weighted dice, this means there would be even less chance of S.I.D.S. happening in their case than on average, not more!”

  I shook my head in wonder at what he had just said, and it was the second time he had used the ‘clear evidence’ term in two days. He must have rehearsed it. Now I was going to have some fun…

  “You checked the environment and found no reason that this was the origin of the deaths?” I asked. “How on earth can you check when you don’t know what to look for? Apart from smoking that is. Could it not be something in the mattress? Some residue from a hidden insect in the woodwork? The father’s after-shave lotion? Since you don’t know, it could be just about anything.”

  I went in for the kill. “Then you say, ‘there is clear evidence of no genetic history of infant deaths’? Is this perjury or are you just incompetent?!”

  “Mr Fielding,” roared the judge, “you will abstain from any such insinuations or insults in this court. I’ve half a mind to hold you in contempt!”

  “My Lord, if you give me time, I can demonstrate that this extraordinary statement has to be untrue, so that it is either one or the other!”

  The judge, furious, glanced to his left. Shaking his head, he seemed to count to three to calm down. “Get on with it then!” he barked.

  “Professor Goodwin,” I said, “you are a man of medicine. If I had had cancer, and seemed to be in remission, you would take blood samples to see if I had any cancerous cells still around?”

  “Yes.”

  “And if you found none, what would you conclude?”

  “That you have been cured.”

  “Might you conclude, if I may paraphrase your words, that there was ‘clear evidence of no disease’?”

  “Yes,” said Goodwin.

  “Did you test all my cells, or only a small sample?”

  “We can only test a sample. It would kill the patient to test every last one of his cells!”

  “So, the cancer cells could still be hiding somewhere that you did not test?” I asked.

  “Er, yes conceivably.”

  “However, you just said ‘clear evidence of no disease’. How can you be so sure? How do you prove a negative? Actually, it’s impossible to prove a negative. Would you not have to test every single last cell of my body in order to prove a negative?”

  “Er…” hesitated Goodwin. “No, we can’t be 100% sure, but statistically there is a very good chance that it’s gone.”

  “May I dare to say that you are mixing up two fundamentally different statistical notions: ‘evidence of no disease’ is not the same thing
as ‘no evidence of disease’?”

  “I see your point,” he conceded. “Well; yes, strictly you should say ‘no evidence of disease’.”

  “Now,” I said, “may we come back to the genetics survey you did of the defendant’s family? I argue you cannot say ‘clear evidence of no genetic history of infant deaths’, you can only say ‘you found no evidence of genetic history of S.I.D.S.’. Do you agree?”

  “Yes, put that way I suppose I have to agree. But it really is splitting hairs…”

  “Splitting hairs?” I said indignantly. “Can you convict someone simply because you found no evidence that can prove their probable innocence? Does not the burden of proof come to the prosecution?”

  “I repeat, we did a thorough search of family history,” said Goodwin defensively.

  “I am a little curious as to your conclusions from this research.” I glanced at my notes. “If some distant family siblings or cousins died in infancy, they are not around to talk about it, are they? Had you considered the bias in your research of what is called ‘silent evidence’ – the evidence to prove the contrary to your hypothesis is no longer there because this same evidence died off?”

  “I repeat we did significant research. Through questionnaires to branches of the family and going through church records.”

  “Asking who? Did all the families boast about, or even know about, infant deaths of siblings, cousins or great aunts? Is there a church record if an infant died before it was christened some 100 years ago? Can you really be sure of your data?”

  “I am satisfied with our research,” he insisted.

  “And how far did you go back?” I asked. “For arguments sake, let’s say this is a gene that increases the probability of a cot death by 100-to-1,000 times the average, which means we would expect between one death in 85 births to one death in 8.5 births – still fortunately a relatively rare event. Might you not have to go back at least ten or more generations to find it at work?”

  I turned to the judge. “May I have a calculator, please, or a calculator app on a smartphone?” One of the ushers pulled out his phone, he glanced at the judge who nodded his assent, then he opened his calculator app and passed the phone to me. I quickly calculated two to the power of ten, “That’s over 1,000 great, great… I don’t know how many great, grandparents to check out over the last 300 years? Did you do all that?”

  “No that would have been impossible,” said Goodwin.

  “How many relatives did you check out?”

  “About twenty.”

  “Going back how many generations?” I asked.

  “Two or three.”

  “So, you checked out only two per cent of my proposed list of over 1,000 possible ancestors and went back only a quarter of the period in time? Was your sample size statistically large enough, Professor? Remember, we are only looking for one rare event, not for a series. I ask you again, could this rare gene conceivably be in the defendant’s, or the defendant’s husband’s, genetic make-up?”

  “Conceivably yes, but I still count it very unlikely,” conceded Goodwin.

  “And finally, could it be the presence of two different genes that trigger S.I.D.S.? Both are necessary, but neither one alone is sufficient – like in our malaria example. Can we exclude this?”

  “No, I suppose we cannot exclude this.”

  “What if one gene came from the mother’s side, and the other gene came from the father’s side? I ask you, even if you did an exhaustive check of a hundred per cent of all the ancestors, would any cot deaths show up if these two genes have not yet met?”

  “Er, no, in that scenario they would not show up any cot deaths,” said Goodwin.

  “So,” I said, “in conclusion, you cannot exclude the possibility of cot deaths being due to genes, and you cannot exclude the possibility of such genes being in the deceased children. Am I right?”

  “I repeat,” said Goodwin, “I have seen plenty of evidence that child abuse goes from one generation to the next, but none to suggest that S.I.D.S. does. However, I also accept that we cannot completely exclude this.”

  “In the light of this discussion, let’s do a thought experiment.” I said. “Assume a case where there is a genetic predisposition to S.I.D.S., but we don’t know which gene. Although the average data from the population would give us an apparent statistical randomness, in reality a family having this gene would have an increased probability of having a baby that dies of S.I.D.S., when compared to the average probability. Correct?”

  “This is your thought experiment, not mine.”

  “Professor Goodwin,” interceded the judge, “please answer the question. Mr Fielding, please repeat.”

  “Is there an increased chance of one family with this gene having S.I.D.S.?” I asked.

  “Yes, in your thought experiment,” said Goodwin.

  “If the gene increased the probability of S.I.D.S. by a factor of, say, 100, what would be the likely probability of one death?”

  “Er, for the Richardson’s household that would be one in 8,500 divided by one hundred, which makes one in 85.”

  “What then would be the probability for a family having this gene suffering two deaths?”

  “85 times 85.” Goodwin used his phone and calculated. “One chance in 7,200.”

  “One chance in 7,200? Is that not 10,000 times smaller than your previously calculated one chance in 72 million?” I asked.

  “Yes,” he said.

  “And if by mischance, the genetic risk multiplied the original probability by as much as 1,000, the probability of two deaths would reduce to one in only 8.5 multiplied by 8.5,” this time I did the calculation on the usher’s phone, “so one in only 72?”

  “Don’t push your luck.”

  “Professor Goodwin, please answer the question!” ordered the judge.

  “Yes, in the extremely unlikely case that the genes multiplied the probability by a thousand, it would be one in 72.”

  “We really are very far from your stipulation of one in 72 million,” I said.

  I carried on. “If there are approximately 800,000 births a year in the UK, and if we continue to use the very high one in 8,500 that you insist applies to the Richrdsons, then we can expect…” I typed in the figures on the phone’s calculator, “about 94 cot deaths a year. OK so far?”

  “As we saw earlier, in reality there would be more nationwide, including those families who are smokers, unemployed or have young mothers” said Goodwin.

  “OK, let’s remain very conservative,” I said. “Let’s assume that a bit more than half of these parents try again – they want a baby, and they will want to forget the death as soon as possible. So, each year there’s around 50 parents having suffered one cot death now trying for another baby. If the cause is genetic, then probably these 50 parents have this gene. Let’s take our lower bound of a one in 85 chance of having another cot death… 50 divided by 85,” I typed in the figures, “gives 0.6 of these babies as dying, already more than one case of double cot death in the UK every two years! If we take the upper bound of one in 8.5 dying, then there are ten times more, that’s six double cot deaths per year! Do you agree with the maths?”

  “OK, but only assuming that you multiply the odds by 100 to 1,000 due to a strong genetic influence. Your assumptions, not mine,” said Goodwin.

  “Now,” I said, “we agreed at the beginning: the accused is not some random person we found on the streets. She is here precisely because she has suffered two cot deaths. According to these corrected statistics, there will be say between one person every two years at least, and six people a year at most, in the UK, in her position of suffering two cot deaths.”

  There was a hushed silence in the courtroom. Even the prosecution barrister was silently shaking his head in dismay.

  “Professor Goodwin, do you agree with these calculations?” I a
sked.

  “Only if the cause were genetic,” he said, “and if your probability assumptions were correct, both of which I doubt.”

  “Can you prove to me that my assumptions are wrong?”

  “Er, no.”

  “And we saw a similar argument if the cause is environmental, didn’t we?”

  “Yes, we did.”

  “And in each case, it’s the same genes and the same environment, isn’t it?”

  “Yes.” Goodwin sounded deflated.

  “So, Professor, now is your chance to redeem yourself,” I said. “We had what appeared to be one of two unlikely events – either double natural death or double murder. It now seems that two natural deaths have a higher probability of occurring then we first thought. In your expert opinion, is the probability of the defendant’s children dying of natural deaths greater or lesser than the probability of her intentionally killing them both?”

  “I’m sorry, but we cannot rule out the marks on the second baby’s face that suggests evidence of smothering. That changes the calculation. There is suspicion of foul play, even if you argue those marks alone could not prove murder. We are no longer talking about your famous 4 million different ‘attempts’ at double cot deaths. Those marks are there only on Mrs Richardson’s baby, not on the other 3,999,999 other births. Only her baby. That makes her a suspect. And once she is suspect, she then only has a one in 72 million chance with my statistics, or a one in 72 chance even with your most optimistic gene altered statistics, of being innocent.” Godwin glared at me in his intellectual triumph.

  I must confess, this stymied me. My mind whirled on what had he just said. Had I underestimated the probabilistic impact of these marks? Was what he said right? I decided, perhaps rather foolhardily, to reason out loud, because my gut feel told me that what he had just said did not add up. “I understand what you are saying, but intuitively it doesn’t make sense. Let me explain my reasoning. Assume that what you call my most optimistic gene altered statistics are correct. We get six or more babies dying each year following two subsequent cot deaths. I would say that there is a good chance that either the mother, father or ambulance crew, someone will try to resuscitate these babies, each time. Our pathologist admitted that resuscitation might well be the cause of the marks. So, at least some out of those six babies might have those marks. Are we going to throw each of those families into jail for murder?”