I went to the flip chart and wrote:
P(2CD | 2D) = 1 / [ 1 + P(2M)/P(2CD)]1
“On the left is the probability of two cot deaths (2CD) having happened, given that the rare event of two deaths (2D) has already happened,” I explained to mostly very puzzled faces, some of which were totally aghast. “That basically means the probability of her being innocent. On the right we see that it is a function of… er, that means depends on, the ratio of P(2M) – which is the probability of two murders having happened – divided by P(2CD) – which is the probability of two cot deaths having happened. Both P(2M) and P(2CD) are incredibly small numbers. But if a double murder is, say, ten times less likely than a double cot death, not unexpectedly this equation gives the chances of her being innocent at about 90% (91% to be exact). At no point in time is her chance of being innocent just simply equal to P(2CD) – the very small stand-alone probability of two cot deaths before any deaths have actually happened, as Professor Goodwin tried to make us believe. In fact, the one to 72 million is a totally irrelevant statistic!”
“You talking bloody double-dutch again, you is,” said Beth.
“That’s enough, Martin,” said Hilary, who probably agreed with Beth. “We’ve got a judge and courtroom waiting for us.”
“If,” said David as he stood up, “you could have convinced me how you derived that equation, Martin, and that you hadn’t just invented it out of thin air, maybe I would have followed your reasoning, and perhaps one or two others here would have as well, and perhaps together we could have convinced enough of the others to acquit. Maybe. But if you hadn’t been on this jury,” he added, “it’s almost sure we would have found her guilty...”
On that sober note, Hilary summoned the jury keeper and we all tramped back into the courtroom.
Back in court, the judge asked our foreperson if we had reached a unanimous verdict on the count of murder of George Richardson.
“Yes, my Lord,” replied Hilary.
“And what is it?”
“Not guilty.” There was an audible sigh of relief around the courtroom.
The judge turned to the defendant. “Mrs Richardson, you are free to go.”
She was sobbing uncontrollably, but she nodded her thanks. Poor woman, I thought, to have lost two sons and then have to put up with the accusation that she had killed them.
Back in the public benches I saw what I assumed to be her husband, and her family, all hugging each other in relief after their months of ordeal.
Some journalists were already beginning to stand up to get out and make their reports. Others were already hard at work thumbing away on their smartphones without moving places.
The judge addressed the court. “Before we end this trial, I would just like to add a couple of comments.” Any people standing up then gingerly sat down again. The judge turned to the public benches where our esteemed professor was now sitting meekly since the lunch break.
“Professor Goodwin, you are a renowned paediatrician, and yet you seem to have ignored some basic scientific principals in your testimony, that even a young engineer, who we are fortunate to have with us today, was able to unwind quite rapidly. I would not go so far as to say that you have committed perjury, but your lack of rigour does in my opinion disqualify you from ever giving testimony in court again. Please take note. Personally, I think you are a disgrace to your honourable profession.
“I am given to understand that your expert testimony has already been used to convict in several other cases of infant mortality. I have no other recourse than to formally request to the Lord Chief Justice that all these cases are reopened and re-evaluated in the light of your statistical and professional mistakes disclosed today.
“Ladies and gentlemen of the jury. You are dismissed and are excused further jury duties.”
Chapter Twelve
As we were gathering our phones and other items from our lockers, Beth could be heard complaining. “I was ’oping for the £130 daily allowance you get for a long trial, that’s more than I get at work. It’s only £65 a day for short trials, multiplied by two flipping days, that’s nuffin’. And all thanks to this guy,” she added cocking her head at me while going off in a huff.
Stephanie and I laughed, but not for long. “Some innocent mother almost got a life sentence, and all she can think about is her indemnities!” said Stephanie.
“Since our time together as jurors has been dramatically cut short by some young fool,” I said, “may I now dare invite you out to dinner later this evening? Or would that cause some boyfriend to be too jealous?” My oh my, I thought, wasn’t I getting a little too bold and a little too direct?
“Well,” she said, “there’s nobody currently qualified to be jealous, although a few boring doctors would like to think otherwise.” (That sounded hopeful!) “But I will only accept on condition you let me eat chips with my steak!”
“Deal!” I laughed.
We agreed on a time, and I said I would send her an SMS with the restaurant’s details once I had found a suitable reservation. We kissed each other goodbye on both cheeks, but I was convinced our mouths were a bit closer than they were supposed to be.
On the way out through the main entrance hall, I bumped into a face that I vaguely knew. Wasn’t it the person who had been sitting in the far-left rear of the courtroom? Yes, but hadn’t I seen that face before somewhere else? He looked a bit like a nightclub bouncer, or maybe an enforcer for some East-end gang. He came over to congratulate me. “Excellent job,” he said, in an American accent. (Not East End then, I thought, maybe New York mafia? Has he been hired by the prosecution to beat me up?) “I suspect you’ve been reading some of my books. My name’s Nassim Taleb.”
My jaw dropped. The Nassim Taleb? The guy who wrote Fooled By Randomness, and even more spectacular, The Black Swan on unexpected events. The guy who predicted the unlikely event of a financial crisis due to ‘fat tails’? My favourite author! I had seen his photograph on the back of his books. In his third book, Antifragile, he explains that some systems actually get stronger thanks to stressors. He also explained that he likes to practice what he preaches, so to stress his body he works out with weights, and now nobody dares to argue with him.
“An honour to meet you, Mr Taleb!” I said shaking his hand. “And an even greater honour to be congratulated by you.” Suddenly I clicked on the judge’s wavering attitudes. “Were you signalling or something to the judge from the back of the court?”
“Yup!” he replied. “The judge knew he was out of his depth when he tried reading your questions. He also knew he had taken an enormous risk in getting you to ask the questions himself. Yesterday evening he asked around for a probability expert. One of his learned friends had read my books and had just attended a conference here in London last week where I was one of the speakers, on ‘Uncertainty in the financial markets’.
“The judge managed to wrangle my mobile phone number from the conference organisers, probably threatened them with a life sentence or something, and he rang me last night. He was happy to hear that I was still in London. He explained the situation to me, read out your written questions and gave me a brief summary of your cross-examination so far. He then asked if he should hire you or fire you. Well, he didn’t actually use that language. Far too polite.”
“And what did you say?” I asked.
“I told him to hang on in there, that they had hit gold by having you in the courtroom,” he replied. “I also suggested that he should throw the expert out on his ass, and that the defence lawyer should be debarred for incompetence in not having better prepared. I probably used slightly different wording as well. I don’t often talk to Lords.”
“He’s not a Lord, although that’s what they call him.”
“You Brits! Why make it simple when you can make it complicated?”
“But thanks for the job reference. How come you were in co
urt?”
“Well the judge made me a proposal,” said Taleb. "He would take me out to the best restaurant in London if I would sit at the back of the court today and signal if you were on the ball or if you were bullshitting. Given the teasers he’d already explained to me, I told him I wouldn’t miss the show for anything. So, I cancelled everything I’d planned to do today, and here I am.
“We’d decided together that my pen pointing in the air meant what you were saying was good; and if it pointed down than you were in deep water. I texted him a photo of me so he would recognise me in court. I can tell you; you did a really good job! Hardly ever did my pen point down. He was just about to fire you for talking about perjury when he saw me actively shaking my tip towards the ceiling. I knew you’d hit gold dust with that fool’s description of ‘evidence of no genetic precedent’. Did you get that from my book The Black Swan?”
“What me? An engineer? I don’t know how to read!” Something else clicked. “You didn’t give him any help with the summing up by any chance?” I asked.
“What? I’m sure that would be unethical,” he said. “But I guess judges can phone whom they want in their meal breaks. So how come you knew all about Bayes’ theorem and the prosecutor’s fallacy?”
“We covered it in our probability maths,” I answered. “To illustrate it we first looked at the probabilities of mammography testing for cancer. The test is judged to have a 90% accuracy rate; so people testing positive wrongly assumed they had a 90% chance of cancer. They didn’t correctly account for the false positives. If I remember correctly, some 0.3% of women risk having breast cancer, so using Bayes’ theorem you can quickly calculate that if a woman tests positive, she still only has a 3% risk of cancer. Which is logical anyway if you understand the notion of a false positive: the mammography eliminates nine out of ten women with no risk; so that if you are not eliminated by the test your risk goes up ten times from 0.3% to 3%. But just like in the prosecutor’s fallacy, the real probability of cancer after a positive mammography, 3%, was often being confused with the 90% accuracy rate.
“Our lecturer then extended this concept to the courtrooms, where the prosecution might make the fallacy of assuming that the probability of a random match of evidence is equal to the probability that the defendant is guilty. In the pre-DNA days, if the police find a certain blood type at the scene of the crime that only 10% of the population have, and the defendant has this blood type, then the prosecution might say he has 90% chance of being guilty; whereas by using Bayes’ theorem – and common sense! – if there are, say, 100 suspects and this guy was pulled out at random, the fact he has the same blood type means he has only a 10% chance of being guilty.”
“You know there is also a similar, almost mirror image misuse of statistics called the defence attorney’s fallacy?” asked Taleb.
“No, I didn’t.”
“It was used by the defence in the O.J. Simpson trial in 1994 for the murder of his ex-wife,” he explained. “The prosecution underlined that there was previous proof of Simpson beating his wife, which could only increase the likelihood of his guilt already derived from other evidence. But the defence used statistics to show that only one out of 2,500 battered women went on to be killed by their battering spouse, therefore the prosecution’s statements were irrelevant to the trial, and should be ignored. ‘Effect given cause’, the probability of the effect, being killed, due to the cause, a battering spouse, only one 2,500; ‘so jury please ignore this’ they said.
“But what the defence cleverly omitted from their calculation was ‘cause given effect’,” he continued. “Simpson’s ex-wife had indeed been murdered; the ‘effect’ was already there, it wasn’t hypothetical. And here the one out of 2,500 statistic was totally irrelevant. Again, using Bayes’ theorem, what counted was, once you’ve got a dead body, what is the ratio of probability of being murdered by a spouse known to beat his wife, divided by the global probability of murder? The answer is a much higher one in 3.5. But that didn’t come out in the trial. That, and some equally erroneous arguments from the defence on the DNA statistics led him to be acquitted.”
“It would seem no statistics should ever be quoted in court without some mathematician to comment on their validity,” I suggested.
“That would certainly be better than nothing,” Taleb agreed, “but many so-called mathematicians also quite often get it totally wrong. Fat tails… need I say any more?”
“I take your point!” I laughed.
“There were mathematicians at this latest financial conference, here in London, falling into the same trap as before,” he said. “Everyone was saying how sub-primes could never happen again; I just said the next financial crisis won’t be sub-primes but some other vastly correlated event that nobody sees coming. I imagine you get the irony of the parallel between the trial you’ve just wound up and the 2008 financial crisis?”
“You mean our dear Professor Goodwin making the very same mistake as all those rating agencies giving AAA risk-free status to those toxic, sub-prime mortgage securities that caused the crisis?” I asked.
“Yup. Got it in one!”
I shook my head in wonder as I took in the enormity of the parallel. “He squared the individual probabilities of cot deaths,” I said thinking out loud, “which assumes statistical independence: one cot death happening has no influence on the probability of another death, he is saying. Unbelievably, all the great banks in the world, and those rating agencies did the same thing before 2008. Because the mortgage securities were made up of loans to thousands and thousands of different houses, they assumed only one or two houses would lose money, but never all the houses at the same time. It can’t happen, they said, just like Goodwin said there couldn’t be two accidental deaths in a row. But just like the Professor, they assumed statistical independence, whereas in reality there was statistical dependence. How could they not see that when a property speculation bubble bursts, all those houses lose value at the same time? All those so-called risk-free securities suddenly all become toxic sub-prime mortgages.”
“But they didn’t see it,” said Taleb. “They missed the correlation – when one mortgage takes a nose drive, there is every risk that the others are doing the same. Just like if one family has a cot death, there is every risk that a second could follow. And right here, did the defence counsel see Goodwin’s evident mistakes? Or did the other members of your jury see it?”
“No, they didn’t,” I confirmed. “In both cases it’s obvious when you think about it. But in both cases people are blind to the obvious with catastrophic consequences. That poor woman could have been sentenced to life in prison.”
“Just like trillions of dollars of people’s hard-earned pension funds were lost in the crisis,” he said. “And just four days ago at that conference, some of the world’s top financial wizards were saying it couldn’t happen again because we learnt our lesson about mortgage securities. I couldn’t believe my ears. It’s almost irrelevant what underlying event caused the crisis, the whole point is that nearly everything is correlated, with knock-on effects that can cause things to happen everywhere, virtually at the same time. They are right to say the next crisis won’t be due to sub-primes, but they are so wrong to say that there won’t be another crisis. It could be anything next time, maybe some unheard-of virus mutating and infecting everybody around the world. Nobody could then say that someone falling ill in Delhi is statistically independent to someone else getting ill in New York! Just think what the stock markets would do if worldwide you couldn’t go to work for fear of being infected!”
“Don’t be such a pessimist!” I said. “I doubt that could happen. You’ll be talking about nuclear annihilation next, with statistically correlated tweets between two egocentric presidents winding each other up, action and reaction, leading to first strike and counter-strike!”
“Better to have the virus,” he said. “Then at least some of us might s
urvive. The human race as a whole could be anti-fragile and survive a virus attack thanks to antibodies, but not to a nuclear attack. But anyway, before someone pushes the wrong button or some deadly virus mutates, it’s me who’ll take you out to dinner – but to the cheapest place I can find – and we’ll talk about some research projects that I’ve got coming up in the USA that might interest you! Are you free tonight?”
“Ah, no. I’ve got a dinner date with a certain young lady.”
“Ah ha! I think I can guess who. You should have seen the way she was looking at you while you were grilling that so-called professor. How about lunch tomorrow then? I only go back to the States at the end of the week.”
“You’re on,” I said. “Which McDonald’s are you taking me to?”
“Oh no, they’re too expensive! Give me your number and I’ll give you a call.” We swapped numbers. What would we do without our mobile phones, I thought?
Taleb ambled off to save the world. Either that, or to write some philosophy, or to calculate the black swan unlikelihood of British trains running on time.
Meanwhile, I walked out of the building to be confronted by a barrage of flashing cameras, journalists and even a few TV news teams. They were of course waiting to interview Sarah Richardson after her ordeal. They would decry the near-miss of a travesty of justice and criticise the police and the Crown Prosecution Service for hounding an innocent woman; neatly forgetting that in their same columns or broadcasts, she had, right up to this very morning, been portrayed as a wicked child killer.
It seemed they were also interested in a young member of the jury who had helped bring down the house of cards. I was about to have a (very) brief two days of fame.
Postscript
The above trial is a highly fictional representation of many real arguments used in, and after, two very real landmark trials on cot deaths, with different mothers as defendants, but with the same expert witness. All names are fictional. The actual statistic quoted by the real-life expert, whom we shall call ‘Professor A’, was ‘one in 73 million for two cot deaths’, and this led directly or indirectly to guilty verdicts in both trials.
The Mathematical Murder of Innocence Page 11