The Mathematical Murder of Innocence

by Michael Carter

  “I really know nothing about waves and oil platforms; I am a doctor,” replied Goodwin dryly. “But I presume you made a mistake in your calculations.”

  “Yes, you are a doctor, sir,” I said. “And from what I see you are definitely not a mathematician!”

  “Careful, Mr Fielding,” said the judge.

  “My Lord,” I replied, “although I do not have any legal training, I am given to understand that an expert witness should only give evidence in his area of expertise. Would you as a judge accept me, an engineer, to give, say, expert medical evidence on infections discovered in pathological post-mortems?”

  The judge visibly bridled. “Under your own admission you do not have any legal training, Mr Fielding, so you are not a qualified lawyer and even less a qualified judge. Please leave your comments on interpreting the law, and who can say what in my courtroom, to me. Meanwhile, get on with it, and explain why your oil platform red herring is relevant to this witness’ testimony!”

  Turning back to Goodwin, I said, “No, our calculations were accurate. It was the data samples that were wrong, or perhaps insufficient. Either the length of time we used to sample wave height was too short, or during that period some random external effect was present – maybe the moon calmed the waters, or some atmospheric condition corrupted the sampling, we will never know.”

  “What’s your point, young man?” Goodwin demanded impatiently. The prosecution was nodding vigorously to mirror Goodwin’s impatience.

  “My point,” I said, “is that your cot death data is not reasonably reliable. Why? Because the chances of a baby’s death happening goes up if we include all natural deaths, and not just those we can’t identify. Because only 44 data points for any given combination of variables is just not enough to determine reliable statistics. Because you mistake description for prescription. Because we see an enormous variability with just a few changes in innocent variables. And maybe it’s not these variables that really change the probability – cot deaths may be due to some other factors that are simply found more often along with, say, unemployed parents or younger mothers; and perhaps these other factors are also present in the Richardson’s household despite the parents’ employment or despite the mother being a bit older. And finally, you have fallen for the ‘ecological fallacy’.”

  “What on earth is that?” asked Goodwin. The expression on the judge’s face seemed to agree with Godwin’s exasperation.

  “It’s when you fall into the trap of assuming that the average of a population applies to each individual in that population. There is an incredible variation of probabilities within any population, your own data proves this. Nobody is average, there will always be individual extremes that average out. It’s not a crime to be different from the average, it’s normal.

  “For example, Professor. Would you cross a river that is four feet deep on average?” There was subdued laughter around the courtroom as what I had said sunk in. “No, I thought not. The Richardson’s household might have only one in 8,500 chances of having a cot death on average; but the actual individual probability for the Richardsons could be very different; your very own data shows that you tweak a few variables and the odds shoot up to one in 200.”

  The judge decided we all needed a break, or at least he needed one. After he had left, we filed out to the jury room.

  My fellow jurors were slightly more muted this morning and seemed more interested in grabbing a quick coffee rather than talking to me. Maybe I had laboured the point too much on averages not applying to individuals, and I doubted whether that had really swayed their prejudices. I guessed ‘the jury was out’ – in all its meanings – as to whether they thought I actually knew what I was talking about.

  When we came back in twenty minutes later, I decided to attack from an even more important angle.

  Chapter Nine

  “Professor Goodwin,” I said, “what are the statistics of a mother murdering her own baby? What is the probability?”

  “I don’t have this exact information.”

  “What, you don’t have this information?” I asked, trying to sound incredulous.

  “No.”

  “In your opinion,” I said, “is it more likely that a baby dies of a cot death, or that a baby is murdered by its mother?”

  “Fortunately, murder is quite rare,” he replied. “So, for a single death it’s more likely to be S.I.D.S. It’s for double deaths that murder becomes the more likely scenario.”

  “What, then, are the statistics of mothers murdering two of their babies in a row, Professor?”

  “I don’t have this information either. But in any case, you can’t simply multiply the probabilities for single murders. You see, a person diagnosed with F.D.I.A. is much more likely to commit murder than someone who is not, and that same higher probability applies to the second murder. The events are statistically dependant.”

  “Statistically dependent?” I repeated with emphasis. “Very interesting and very important concept. We will come back to this. But in any case, you couldn’t multiply the probabilities even if you wanted to, because you don’t have them in the first place!”

  I caught another warning glance from the judge.

  I moved on. “Have you heard of the ‘prosecutor’s fallacy’, Professor Goodwin?”

  “No, I don’t believe I have. But doubtless you are going to enlighten me…” he said, rolling his eyes upwards. It seemed to me that Mr Scott was doing the same.

  “It’s when you confuse ‘cause given effect’ with ‘effect given cause’.”

  Goodwin was shaking his head. “As I said when the judge asked me that question yesterday, it sounds like semantics to me.”

  “Oh no,” I shot back. “It’s not semantics. We know the ‘effect’ – two of Mrs Richardson’s children have died. The court’s role is to determine the cause of this effect, given that this rare event has actually already happened. However unlikely it was, it has happened. We should now analyse all the causes: natural, murder; maybe something else. And we should compare the probabilities of each cause to see which is the most likely, even if all of them have only a small chance of happening. It’s the relative probabilities that count, not the absolute probabilities. For arguments sake, let’s say double cot deaths have one chance in a million of happening, but double murder has one chance in ten million: they are both extremely unlikely. But the unlikely has happened, and by comparing the two probabilities, we see that natural deaths are, in this example, more likely than murder. ‘Cause given effect’.

  “But you, Professor,” I continued, “have only been analysing one cause, cot deaths, for the purpose of eliminating it. You come up with a very small probability. (I believe you calculated wrongly, but we’ll come back to that.) You calculated a probability so small that you immediately jumped to the conclusion that it is murder without even trying to calculate and compare with the probability of murder. ‘Effect given cause’. The ‘effect’ – double deaths – given the ‘cause’ – S.I.D.S. – is you think so unlikely to happen that you unilaterally eliminate it as a possibility.”

  “I’m not really sure I fully follow you,” said Goodwin trying to sound as confused as possible.

  “I think you do,” I said.

  I tried a parallel example. “Surely as an experienced medical researcher you know that all hypotheses have to be tested against a second null hypothesis?”

  “Of course, I do believe that I know more about medical research than you do, young man!”

  “But where are the two hypotheses, Professor? You need to compare the probability of an unlikely event of two cot deaths with the null hypothesis of the probability of another unlikely event of double murder? I only see one set of analysis. You yourself surprised me just now by saying you did not even know the probabilities of murder.

  “Only if the probability of double cot deaths is much, much less l
ikely than the probability of double murder,” I continued, “can you then say that there is a statistical probability of murder. But you didn’t even analyse the probability of murder. You willingly admit that a single cot death is more likely than a mother murdering one child. However, then you assume that double cot deaths are so unlikely, that you simply jump to the conclusion that it therefore had to double be murder! Not very scientific…”

  The prosecution was jumping to his feet. “I really cannot accept that these comments go to the witness’ character!”

  “Mr Fielding,” interrupted the judge. “I will not have you making any insinuations against the esteemed professor. I must confess you have also completely lost me in what seem to be useless semantics. I think we will stop your…” The judge stopped in mid-sentence. He seemed to be glaring at someone in the public gallery over to his far left. A look of resignation came over his face. “Oh, very well. On second thoughts, you may continue your questioning. But please ask simple questions, stick to facts and avoid giving your opinion.”

  “My apologies, sir,” I said. “I will try to rephrase and simplify. Professor Goodwin. When testing a new drug in clinical trials, do you check for the possible effects of the drug against a null hypothesis of the same effects happening by chance. That is do you administer the drug to half the test population and administer a placebo to the other half?”

  “Yes, that is the normal procedure.”

  “In this case did you check and compare the probabilities of both hypotheses? Natural deaths and murder?”

  “Er, no. The probability of natural deaths is so small that it wasn’t necessary. It was obviously murder,” affirmed Goodwin.

  “‘It wasn’t necessary’,” I repeated incredulously. “You just waived standard scientific protocol of analysing the two probabilities. ‘Obviously’, you say. Obvious to you, perhaps? But not to me. Did you not just admit that murder, let alone double murder, was also a very rare event?”

  “Er, yes, I did,” he admitted.

  “And did we not conclude, even using your statistics – which I am going to question in a minute – that due to the sheer number of births there is probably at least one set of double cot deaths per year happening by accident somewhere in the world?”

  “Yes, we did. In the world, perhaps; but not in Britain,” Goodwin added rather churlishly.

  I paused to review my notes from the small hours of the night. “Let us come to what is going to be the most important part of our analysis together. The likelihood of a double event. Professor Goodwin, just now you said that someone who had already committed murder was more likely than someone else to commit murder again?”

  “Yes indeed, a person diagnosed with F.D.I.A. is much more likely to commit murder than someone who is not, and that same high probability applies to the second murder.”

  “So you confirm what you said earlier, the events are statistically dependant?” I asked.

  “Yes,” he said nodding.

  “So, for the double murder possibility, you insist on it not being two independent random events. That once a murderer, there is much more chance of again being a murderer?”

  “Yes.”

  “Now, for cot deaths, or S.I.D.S., you squared the individual probabilities. This implies statistical independence. So, I conclude that you know exactly the cause of cot deaths?”

  “Young man,” he protested. “I never ever said such a thing!”

  Mr Scott was on his feet again. “May I suggest to the judge that we stop this charade? It is evident that Mr Fielding has not followed one iota of Professor Goodwin’s previous testimony!”

  “Mr Fielding,” said the judge, “if you are going to deform the testimony of the witness, I do indeed suggest we stop now.”

  “Please let me continue, my Lord. I will show how Professor Goodwin has indeed contradicted himself.”

  Judge Braithwaite looked around the court room. Suddenly he arched his eyebrows. He added wearily, “OK just a few more questions, but please be careful, Mr Fielding.”

  I decided to rewind a bit for clarity. “Professor, for double murders you say the events are statistically dependant. Once the first has happened, it is much more likely that the second will happen, because the first event proves the person has a higher risk of doing it again?”

  “Yes, that is exactly what I said.”

  “But you say we do not know the cause of S.I.D.S.?”

  “No, we do not.”

  I turned to the judge. “My Lord, we will now go through the arguments that if we do not know the cause of S.I.D.S., then it is incorrect to assume statistical independence such as Professor Goodwin has done by squaring the individual probabilities.”

  “I’m not sure I follow you,” growled the judge. “But get on with it.”

  “Professor Goodwin,” I said, “when you calculated your probability, you multiplied one in 8,500 by one in 8,500. You used exactly the same probability for the second cot death as you did for the first cot death. Why?”

  “We do this because cot deaths appear randomly in the population.” He now turned to the judge. “My Lord, contrary to Mr Fielding’s assertions, when events are random, there is indeed statistical independence, and so the correct calculation is to square the individual probability. If you roll a dice and you randomly get a six, when you roll it again there is no more chance of having a six than of having any other number. It’s random both times. In this simple example the chance of two sixes in a row is one out of six squared, that is one chance out of 36. About the same probability as juror number six being right twice in a row!” There was a chuckle around the courtroom at this jibe at my expense.

  “Now, now, Professor, that will do,” said the judge. “I’ll give you my indulgence, seeing as you have yourself been the victim of the same from our rather too enthusiastic young juror here. Mr Fielding, even I can understand that the chances of two sixes in a row is one in six squared. Does that answer now satisfy you?”

  “Not really, sir. Professor, you are specifically saying that if you have one cot death in the family, that doesn’t create any more probability of having a second cot death in the same family?”

  “That is exactly what I am saying.”

  “And you say this because a cot death is totally random. It’s like playing Russian roulette with a gun with 8,500 chambers, and only one of the chambers has a bullet in it? Each time you pull the trigger you have a one in 8,500 chance of a death.”

  “Yes, that explains it very well.”

  “Are there no external factors that contributed to the first cot death that then contribute to the second cot death?”

  “No, I repeat, the data suggests it is purely random.”

  I’m glad you insist on that, I thought. I looked down at my notes. Yes, time to trip him up with a new argument. “But you said the one in 8,500 were the statistics for non-smoking families? Why did you use those statistics in particular?”

  “Because the Richardsons are non-smokers. I wanted to be as close as possible to the true family environment,” he answered.

  “So, smoking increases the chances of S.I.D.S.?”

  “Yes.”

  “I’m sorry,” I said without feeling it in the least, “but in that case, you are contradicting yourself. It can’t be completely random if smoking increases the odds.”

  “If you walk through a minefield, just like you are doing right now young man,” bridled Goodwin, “everyone knows that you have a higher chance of being killed. Please don’t play the naive fool with me; we all know that smoking is a serious health risk.”

  “But when was smoking finally recognised as being dangerous?”

  “Right back in the 1950s, if I remember correctly.”

  “But smoking came to Europe from South America in the 1500s,” I pointed out. “For four centuries nobody knew it was dangerous.
Is it possible that, say, in 1900 – not so long ago really – after a heavy smoker has died of lung cancer, some eminent medical professor might have said, ‘He died of some random arbitrary disease called cancer’?”

  “Yes, I suppose so. But what are you trying to get at?” Goodwin sounded righteously irritated. “I told you I used the statistics of non-smoking families.”

  “My point is twofold. First, you assume that there are only random causes for S.I.D.S. Then you contradict yourself by saying smoking is a contributary cause.”

  “Which we agree seems pretty obvious.”

  “And I argue that for many centuries people were smoking in Europe without knowing it caused any problems. Only relatively recently has smoking been identified as a cause of lung cancer. Until then the causes of cancer appeared random, because people hadn’t yet identified any of the causes.

  “Now for S.I.D.S.,” I continued, “you yourself admit that parents’ smoking contributes – that is one cause you have so far identified. But then you say all the other reasons for S.I.D.S. are totally random. I disagree. It is simply that you haven’t yet identified the other causes, so they appear random. In statistics when you see a result that appears random because you do not know the causes, you should not make the mistake of then assuming that the causes are also random. The causes, once known, are probably anything but random.”

  Goodwin chose not to react to this analysis.

  I glanced down at my notes. “I would like to pursue another angle to the same idea. Have you ever had malaria?”