Vsauce! Kevin here -- and I want to go over a real-life math mistake that ended up actually
making a murderer.
Our top story tonight: she seemed to be the woman with everything - a good job, a lovely
house, and the beginnings of a young family. But tonight, the life of Cheshire solicitor
Sally Clark is in tatters after she was convicted of killing her two baby sons. Two children,
both dead before their first birthday. Sally Clark, the mother, is arrested. Charged with
murder.
What happens when two improbable tragedies happen to the same person? How do you calculate
odds to differentiate unbelievably bad luck from murder? And what happens when you make
a conditional probability error so severe that it decimates countless lives?
I’ve talked about a lot of paradoxes and math problems that, honestly, are kind of…
theoretical. They’re REAL, but the situations involving them don’t come up a whole lot.
You’re not THAT likely to be a game show contestant or have to figure out whether you
should go first in Russian Roulette.
And it doesn’t really matter whether you can play a dot game with perfect efficiency.
It’s awesome to know how it works and why, but it’s ultimately not life or death.
But... some math is life or death.
On February 23, 1998, Sally and Steve Clark were arrested and charged with murder a month
after their infant son was found dead. He was only 2 months old. Sally and Steve claimed
that he had died naturally, a case of Sudden Infant Death Syndrome -- also called ‘crib
death’ or ‘cot death’ -- that couldn’t possibly have been prevented. The baby had
physical trauma including a cracked rib, which his parents claimed were from resuscitation
efforts.
So… You can see why the police would investigate. The real problem -- the thing that makes the
case notable, and where the math comes in -- is that it wasn’t the first time one
of the Clarks’ infants had been found dead.
Almost the exact same thing happened with their first child about a year earlier.
So what are the odds that an incredibly rare tragedy would happen twice in the same family,
like… the horrible opposite of winning the lottery more than once? And the major question
is, what are the odds of that compared to the odds that one or both of those children
were murdered?
Well, there was an expert for exactly this sort of thing. Sir Roy Meadow was a pediatrician
who built a fantastic reputation on his 1977 paper outlining Munchausen Syndrome by Proxy
-- where a caregiver secretly makes someone sick so they’re permanently dependent, or
sometimes just to be seen as heroic and selfless by making the sacrifice to care for them.
By the early 1990’s, Meadow had become a recognized authority on child abuse. He literally
wrote the book on it -- “The ABCs of Child Abuse” -- and testified at dozens of trials
involving allegedly abusive parents.
It’s how we got “Meadow’s Law,” the maxim Sir Roy used to summarize his attitudes
toward abuse cases: “One is a tragedy, two is suspicious and three is murder unless there
is proof to the contrary."
And that’s where we find our first red flag -- he lifted his “law” from the James
Bond villain Goldfinger.
By now, we’ve discarded Meadow’s Law, and Sir Roy Meadow’s legacy is completely
tarnished -- because he either didn’t know conditional probability or he didn’t care,
and Sally Clark paid the price.
Here’s how conditional probability actually works. It’s figuring out the chances of
something happening given that something else has already occurred. It’s the probability
of A, then B. Sometimes that’s a succession of totally unrelated events, like coin flips.
Whether I get a YES or a NO on the second coin flip doesn’t depend on what I got the
first time. It’s the odds of THIS times the odds of THAT, and we get a value for the
chances of both independent things happening together despite neither one having ANYTHING
to do with the other.
Let’s take cards. The odds of drawing a heart are 1 in 4. Odds of drawing a second
heart? 12 in 51… cuz we just took one out of the deck. Two hearts in a row is just ¼
* 12/51 = 12 in 204. That’s easy -- and thinking that independent events somehow influence
each other when they don’t? That is the famous gambler’s fallacy. You don’t wanna
fall for that.
But sometimes two events are related in some way, and figuring out the odds of both of
them happening together gets trickier.
Like, the average chance of getting into a car accident is around 1 in 350 for every
1,000 miles you drive. So, what are the odds of getting into two accidents in two months,
assuming you drive 1,000 miles a month? Do you just do some multiplication for two independent
events, 1/350 * 1/350 to get 1 in 122,500?
Ehhhh… You COULD do that, and that’s an okay place to start -- but what if you’re
the worst driver in the world? What if you’ve only been driving for a couple of weeks? What
if it’s winter and everything is icy? Is there a difference between driving in Los
Angeles and… Wyoming? Or what if you’re a Fast & Furious driver and just had the unluckiest
couple of months of your life? As the information changes, the math changes.
And this is all common sense -- so how did Roy Meadow completely mess this up?
During the trial, Meadow cited a report that established the chances of an infant dying
from Sudden Infant Death Syndrome using all sorts of different risk factors, like the
mother being young or being a smoker. Sally Clark was a healthy woman with a good standard
of living -- she was a lawyer -- so the chances of someone in her circumstances suffering
a SIDS tragedy was calculated at 1 in 8,543 -- and this is where Meadow’s math went
horribly wrong.
He just multiplied 1/8,543 by 1/8,543 to get the chances of the two independent events
occurring twice for the same mother -- which comes out to 1 in 73 million. Given the United
Kingdom’s birthrate, Meadow said it’s the sort of coincidence that would happen
once every century. He literally treated Sally Clark’s deceased children like playing cards
or coin flips instead of human beings living in complex circumstances.
Here’s a question: What does 1 in 73 million LOOK like? In 2014, Tim Urban created an image
of one million dots. So here’s 73 of those. And as of 2021, only 19 of the 235 countries
in the world have a population over 73 million.
You could make an entire documentary series out of all the medical evidence to consider
in this trial -- but Meadow’s testimony reduced all of the complexities into something
the court could easily understand: Two accidents like this are so statistically improbable
that it makes way more sense to think she killed her kids.
That type of faulty logic actually has a name -- the Prosecutor’s Fallacy -- which is
misusing statistics to create the impression that the accused has to have done the crime.
Sometimes that fallacy’s used by accident, sometimes it’s used on purpose.
But in this case, you can’t even give Meadow the benefit of the doubt and say he misspoke
or just wasn’t thinking -- because he even compared the odds of both Clark children dying
naturally to someone betting successfully on 80-to-1 longshot horses for 4 races in
a row.
The problem is saying something only has a 1 in 73 million chance of occurring naturally
suggests that it’s way more likely the explanation is simpler. For Sally Clark, that meant being
guilty of infanticide.
What’s truly weird to me about this whole story is, people don’t ALWAYS assume the
worst just because the odds of something happening are crazy. When your Grandma Ethel hits a
1 in a million jackpot on a casino slot machine, you think, “Oh my god, yay, she is INSANELY
lucky!” We DON'T immediately suspect her of being a cheater. We think it's amazing,
not utterly impossible. But we think about crimes differently -- someone’s been charged
and the suspicion is baked in.
If you’re a juror, isn’t 1 in 73 million WAY way beyond a reasonable doubt?
Beyond defective math, there was a lot going on here -- like the Clarks’ baby wearing
an apnea monitor that would go off if the baby stopped breathing. That was standard
practice after a family suffered a SIDS death in the past. The child’s monitor kept going
off in the months prior to his death, and public health officials said the unit was
faulty and had it replaced. Did that unit actually malfunction? Is there a chance they
misread warning signs?
As we add layer upon layer of complexity, simple statistics are even more appealing
as a clean explanation for a difficult problem. It’s not hard to see that an entire trial’s
worth of facts could boil down to one question: Is it the 1 in 73 million chance that Sally
Clark was the unluckiest mother in Britain, or one of the other 72,999,999 outcomes? And
whether or not you’re misled by the implications of that statistic makes all the difference.
Sally Clark was convicted of murder by a vote of 10-2, and she was given a mandatory life
sentence. Newspapers reported that she was guilty of killing her babies. They called
her Baby Killer, Baby Murderer, and Killer Mum. They described her as a “Tortured Soul.”
Her husband Steve insisted that she was innocent. Her appeal was dismissed because the court
claimed the statistical errors were quote, “[in]capable of affecting the safety of
the convictions.” Roy Meadow backed them up, saying the time spent on an appeal was
quote, “wasted.” But the Royal Statistical Society blasted Meadow and the court for butchering
such basic math stating:
“...there is a real possibility that without proper guidance, and well-informed presentation,
frequency estimates presented in court could be misinterpreted by the jury in ways that
are very prejudicial to defendants.
Society does not tolerate doctors making serious clinical errors because it is widely understood
that such errors could mean the difference between
life and death. The case of R v. Sally Clark is one
example of a medical expert witness making a serious statistical error, one which may
have had a profound effect on the outcome of the case.”
Sally Clark appealed again. Not only was the statistical evidence flawed, but her lawyers
discovered medical documents showing her son actually died from a serious bacterial infection.
The math was wrong and there was now a rock-solid medical explanation for his natural death.
5 years after the trial, analysis in the journal Paediatric and Perinatal Epidemiology showed
that as rare as double SIDS is, double infanticide is even more rare -- making double SIDS anywhere
from 4.5 to 9 times more likely than double infant murder.
Sally Clark was released in 2003 -- and between the grief of losing two children in tragic
circumstances, being wrongly imprisoned for 3 years, and having the entire United Kingdom
think she was the worst mother in the world, she died of alcohol poisoning just 4 years
later. She left behind her husband and their third child.
The fallout was immense. The British General Medical Council removed Roy Meadow from the
British Medical Register. Cases in which Meadow testified were revisited, and the High Court
ruled that expert witnesses were not immune to disciplinary action. He was barred from
being an expert witness, endured a series of hearings about professional misconduct,
and by 2009 he was not licensed to practice medicine.
Roy Meadow should have known how basic probability worked. You know and I know -- so we should
expect a highly-educated medical professional willing to testify in court to know, too.
But he also blew it on the science -- his misleading statistics dismissed the possibility
of any cause of death that couldn’t currently be identified, like an unknown genetic defect.
Instead, he multiplied two fractions because it was easy. And most of the courtroom bought
it.
Whether people think math is boring or it’s a waste of time to think through theoretical
concepts, the fact is: we learn the theoretical so that we know how to solve problems in reality
-- and sometimes to avoid terrible tragedies.
It’s why we have to understand.
Look. There’s an old conditional probability joke about a statistician who said that he
always flies with a bomb in his suitcase. When someone incredulously asked him why,
he said, “Because the odds of TWO people having bombs on the same flight are astronomical!”
If you get why that joke is funny, you get conditional probability -- and if that courtroom
knew how it worked, Sally Clark would probably be alive today.
And as always, thanks for watching.
Today is not a victory. We are not victorious. There are no winners here. We have all lost.
We simply feel relief that our nightmare is finally at an end.