It All Adds Up

It all comes down to the numbers - at least it does according to Dr. Ric Crossman. You’d be amazed at what can be explained by fractions, decimals, percentages and statistics; let Dr. Ric guide you…

I’d like to start this piece with a quick story. Once upon a time there was a woman named Sally, who bought a lottery ticket. She chose her numbers at random, not really even thinking about it, and then went about her day. Come the weekend, she isn’t even really paying attention to the television as the numbers are read out.

Until she realises the first four numbers are all on her ticket.

Excited, she puts aside her magazine, and stares at the screen. A fifth number appears, also one of hers. Her heart forgets its rhythm. She wonders if she’s still breathing. The last number appears...

It’s hers!

Within moments she has made the call, and claimed her winnings. This will change everything, she thinks. She can buy a new car; a new house. Buy new toys for her newborn baby.

Then the knock at the door comes. It’s the police. “Will you come with us?” they ask, polite but firm. “There are some questions we’d like to run by you.”

Fast forward a few weeks and Sally sits in a courtroom. Her husband is there, wrapped in stunned shock, but her son is nowhere to be seen. The charge against her is fraud, as the prosecution is only too happy to remind the jury time and time again.

And the proof upon which the case against her rests? She must have cheated, because the odds of actually accidentally guessing all six numbers are so astronomically unlikely.

Sounds ludicrous, right? Like something Kafka might come up with on a day when he wasn’t trying all that hard. And yet that’s exactly what happened to Sally Clark in 1999.

Except no, it wasn’t. What happened to Sally was an event a little less likely than winning the lottery, and infinitely less welcome.

In 1996 Sally lost her first child to Sudden Infant Death Syndrome. That’s a loss so horrible and meaningless and personal that it almost seems morbid to attach a figure to it, but roughly speaking for babies in similar conditions to Sally’s son it claims the lives of one in every eight and a half thousand.

As if this wasn’t enough, as though losing a child so early into their precious, unique life wasn’t sufficiently unbearable, Sally lost her second child to the same condition just two years later.

SIDS is idiopathic: that is, its cause is unknown. More to the point, it is so little-understood that it would be theoretically possible to murder a baby by smothering and plausibly claim that SIDS was the culprit.

That’s exactly what Sally found herself being accused of. If there is a one in 8.5 thousand chance of a child dying from SIDS, argued the prosecution, then the chance of two babies losing their lives to it, one after another, is a minuscule one in 73 million. That’s more than five times less likely than winning the lottery from a single ticket.

So she must have done it, right? The majority of the jury certainly thought so, and sentenced Sally to life accordingly.

In truth, though, this makes no sense, any more than it would for the lottery story. After all, the chance you’ll win the lottery this week is infinitesimal. The chance someone will win it is actually pretty decent.

The same, unfortunately, is true of Sally’s tragic loss. How many families in this country have more than two children? My parents had three. So did both sets of my grandparents, along with my aunt and two of my three uncles. All of their children, thankfully, made it through their cot-bound years. But the chance of at least one of those six families suffering Sally’s heart-wrenching loss isn’t one in seventy three million, but about one in thirteen million. We can take this further. There are 17.1 million families in this country, with an average number of 1.8 children. Roughly (very, very roughly), that’s 13.7 million families with more than one child, putting the chance that at least one family in the UK lost two children to SIDS at seventeen percent, or about one in six. We’ve gone from winning the lottery to rolling a seven in craps. And all of this without considering the possibility that SIDS might have some genetic component, making one in 17 million far too small an expression of the true chance.

The (comparatively) happy coda to this unpleasant tale is that the mistake was eventually realised, and Sally freed, after spending three years behind bars (sadly, she passed away four years ago). But the fundamental mistake behind her conviction, the failure to consider the context in which something is brought to your attention (often referred to as “Prosecutor’s Fallacy”, a term used to cover two very different misunderstandings of probability), is very hard to shift.

Not surprisingly, the media are no better at processing this than the courts are, if indeed they give any thought to such things at all. One recent and infamous example was pulled from the effluence by the remarkably unpleasant Jan Moir, who wrote following the tragic death of Steven Gately that his passing that “[S]trikes another blow to the happy-ever-after myth of civil partnerships”.

Well, maybe it does, if you blind yourself to the fact that amongst the hundreds of thousands of gay relationships in the world, some of them are going to end badly (Moir mentioned Heath Ledger in the same article, but she didn’t attempt to argue his premature death proved heterosexual marriage is circling the drain or, for that matter, that it must mean Christopher Nolan is shit). Gately’s death proves nothing more than that amongst all the people who died in 2009, some of them will be gay, and some of those will be famous.

We can find an even more recent example in the baffling thought processes of Ben Stein, who argued this week that Dominique Strauss-Kahn is unlikely to be guilty because amongst the ranks of convicted rapists, you don’t tend to find too many who know their Friedman from their floating currency. This, in fact, is the other form Prosecutor’s Fallacy takes. Stein is confusing the percentage of rapists who then turn out to be economists with the percentage of economists who then turn out to be rapists. Hardly anyone is an economist, so any given rapist is unlikely to know too much about present values and income rates. But you can swap out “rapist” in that last sentence for “French” and it still follows. I’m not saying French are more ignorant of economics than anyone else, obviously. I’m just pointing out that saying “There are hardly any Frenchmen who are economists, so is it really likely that Strauss-Kahn is French?” is no more stupid than Stein’s argument. It’s a great deal less sexist, though.

If nothing else, Sally Clark’s story – along with the ugly, humiliating flailing of Moir and Stein - should stand as a lesson to anyone who wants to draw conclusions from a single event. Observing it and then working back to the cause is a tricky business, and you need to be very, very careful if you want to give it a try. The world is complicated, and the machinery that works beneath and within us all both wondrous and endlessly strange. We might use the phrase “Prosecutor’s Fallacy” to describe those who make ham-fisted attempts to pull apart those mechanisms, but really, whether you’re acting as prosecution is beside the point.

All that really matters is that you're a dick.