A VICTIM OF PROBABILITY

Planning a crime; police interviews; alien abduction; psychology in the courtroom; sentencing; probability and reasoning

Let’s imagine a desperate criminal and his passage through the criminal-justice system. Invisibly, undetected, numbers follow him from hideout to courtroom to prison cell. Invisible and undetected - but not by us, not by subscribers to the great Crime & Psychology newsletter! Since our villain needs to be desperate – as you’ll see, he ends up sentenced to hang – he’ll require an appropriately blood-curdling name. Let’s call him Selwyn.

Selwyn got involved in, let’s say, a heist. It did not go smoothly. A passer-by was injured or killed. It may have been an accident, but Selwyn takes no comfort from that. The fact doesn’t make him feel any less aggrieved. Selwyn - so he’ll tell you - never wanted to be involved in the first place. It was a spur-of-the-moment decision. The plausible rogue who organised the heist managed to dazzle him with an unfair argument rooted in probability theory. After a lengthy cooling-off period, Selwyn came to recognise the mistake in his associate’s reasoning, but by then it was too late.

The associate needs a blood-curdling name, too, perhaps even more so than Selwyn. We’ll call him Kevin.

Kevin - let’s be honest - is a bit of a loser. A repetitive loser, in fact, a career loser, a loss-magnet. This was Kevin’s sixth time planning a heist and his sixth arrest. You’d imagine that Selwyn would have recognised that five-heist losing streak for what it was, but no. Kevin actually contrived to use it as an enticement. ‘What is the probability,’ he asked, ‘what is the probability that I’d get arrested six times in a row?’

‘Five times is bad enough,’ Selwyn wisely observed. The best guide to future behaviour is past behaviour. Selwyn suspected Kevin might never achieve his ambition of becoming a criminal mastermind.

‘But six times?’ Kevin asked. ‘Six times? Never gonna happen. It’s practically impossible.’ Kevin is the type who carts his own bomb through airport security, on the grounds that there is little chance of there being two bombs on any one plane. That, too, is ‘practically impossible’. Kevin is also the type who thinks that the phrase ‘practically impossible’ makes sense.

And so Selwyn got suckered in. ‘Suckered’ is the word, because our friend has fallen for the so-called ‘gambler’s fallacy’. The occurrence of one in a string of independent events (in this example, heists) does not bear on the probability of the occurrence of the others. It’s a difficult concept to grasp when you put it like that. It’s easier if we remember why it’s called the gambler’s fallacy. It afflicts lots of people who are less well-off now than they might have been. Say a punter goes to the race-track to watch ten horse races. The punter notices that, in three consecutive races, the favourite has lost. That hardly ever happens. He reckons the probability of the favourite winning the fourth race must now be relatively high. After all, what’s the probability of the favourite failing to win four consecutive races? Never gonna happen. It’s practically impossible.

Our gambler forgets that, if the probability of the favourite winning a race is, say, 1 in 3, it remains so no matter what happened in those previous, unrelated, races that those previous, unrelated favourites failed to win. Think about coin tosses. The probability of throwing a Head when you toss a fair coin remains 50%, no matter what happened last time, or how many consecutive times Tails has come up.

In fact, Selwyn is doubly unfortunate. He doesn’t know it and perhaps he’s happiest that way. The poor chap was forced to drop out of Statistics class early (he got sent to prison). Had he stayed he would have understood that the five previous arrests are not even the truly independent events of the sort that Kevin implies. They have one thing in common - Kevin himself: fivefold-failure Kevin. That on its own should have been enough to tell Selwyn that his friend simply wasn’t very talented at the whole crime game. Kevin is a no-good criminal in more ways than one.

Let’s go back to the planning stages. Kevin and Selwyn discussed every detail carefully. They made sure to calculate and recalculate how much loot they stood to score. The more they talked, the more convinced Selwyn became. That was a pity for Selwyn, now a victim of the the so-called ‘planning fallacy’.

We are usually too optimistic when it comes to forecasting our own success. The more effort we devote to our plans, the more difficult it becomes to recognise how unrealistic they may actually be. One wise rule of thumb in life is to abandon any project at the point when its projected outcome appears close to a best-case scenario. Another is always to consult the relevant statistics. Have previous attempts been successful (Kevin’s haven’t)?[i] Failures in the past usually imply failure in the future: your future.

Before long, unsurprised Selwyn finds himself in the local nick. He is ‘helping police with their enquiries’. The officers have separated him from Kevin. Neither man can communicate with the other. Selfishly, each wants only to secure the minimum possible sentence for himself. The officers present the two criminals with a dilemma. If both of you stay mum, they say, you’ll likely go free in a few months, after an investigation. If both confess, we will recommend that each of you gets exactly five years. But if one of you confesses and the other does not, we’ll make sure the former goes free while for the latter we’ll recommend the maximum twenty-year stretch.

This is called the Prisoner’s Dilemma. It’s used in Game Theory (which we shan’t go into here) to demonstrate that when individuals act in their own self-interest, the result can actually be worse for them than if they co-operate.

The problem for Selwyn is that he cannot know what Kevin’s going to do. What he does know is this: whether or not Kevin confesses, he himself profits most by confessing. If Kevin confesses too, the police will recommend five years; if he does not, Selwyn will go free. In either case, he’ll avoid the maximum sentence. Kevin – though, as we’ve established, not a mastermind - will surely reach the same conclusion. Hence it appears to be in both of our friends’ interests to confess. The strange thing is that both of them would nevertheless be better off using the irrational strategy of omerta. That would see them free in a few short months.

Selwyn knows all about Game Theory (he reads a lot of behavioural economics in his spare time, of which he has plenty) but he also knows a little bit about Kevin. After the latest failure, he’s more convinced than ever that Kevin is not to be trusted. Both men soon wind up in the dock, facing a judge not widely celebrated for his leniency.

Before the trial, Selwyn’s attorney floats a proposition. He suggests that Selwyn confess. The attorney is absolutely sure that he can plea-bargain a ten-year sentence. If however they go to trial…well, the evidence seems equivocal. The attorney puts the probability of winning the case at exactly 50%. If he does win, of course, Selwyn will serve zero time in prison. But should he lose, Selwyn will surely get pasted with that twenty-year sentence. What should he do?

Selwyn doesn’t ponder long. He has taken enough Statistics classes to know about the concept of expected value and enough Psychology classes to know about risk aversion. (He has had a lot of time on his hands in the past and all his favourite prisons run night-classes.) Selwyn knows that the more a person has of something, the less value they place on getting more of it. A box of Uncle Ben’s, after all, means nothing to a rice magnate, but to a hobo it’s dinner.

Put it this way: If your wealth is a pile of stones, you use the smallest stones at the top and the biggest ones for the foundation. Any stone you remove from the top will be bigger than any stone you could replace it with. The pain you’ll get from removing a stone will be greater than the pleasure you could gain from adding another one.[ii]

Economists would say that the expected value of the attorney’s two propositions is identical. Selwyn must accept either a 10-year sentence or a combined 50% probability of a 20-year one and a 50% probability of none at all. Quick mental arithmetic will confirm that (50% x 20 years) + (50% x 0 years) = 10 years. The outcomes are identical.

Selwyn, a risk-averse character, is not enticed. He absolutely does not want to lose those extra ten years (not even the prospect of prison night-classes is tempting enough). The pain he’d get from a twenty-year sentence would be greater than the pleasure he’d gain from receiving no sentence at all.

Our friend therefore opts for the plea-bargain. But – surprise! - the judge does not do as the attorney predicted. In fact, he sentences Selwyn to be hanged by the neck until he is dead.

Why this unexpected decision? It’s easily explained. Over lunch, the judge was chatting (altogether inappropriately) with an expert witness whose specialises in statistics. Sadly, after too many glasses of wine, the statistician commits the ‘error of inversion’, which is sometimes called the prosecutor’s fallacy. We touched on it briefly in last Sunday’s e-mail.

What this means, technically, is that, given two events, A and B, which are somehow linked, the probability that B will happen if A happens is different from the probability that A will happen if B does.[iii]

I agree. That does sound confusing. Here’s a more colourful way to put it. A small yet surprising number of people suffer from a psychological syndrome known as sleep paralysis. Sometimes, when they wake up, they are literally unable to move. An even smaller, yet still surprising, number of people believe that they are regularly abducted by aliens. One way to tell that it’s about to happen is this: a small ‘grey’ with big eyes and a little mouth appears in your bedroom at night and zaps you with a paralysing ray gun

The probability that you will wake up paralysed IF there is an alien in your bedroom with a paralysing ray gun is clearly very high indeed. The probability that there is an alien with a paralysing ray gun in your bedroom IF you wake up paralysed is clearly very much lower.

Selwyn was caught running down the high street with a sack over his shoulder. Lettering on the sack said swag. We can agree that this doesn’t look good for Selwyn.

Th statistician says this to the judge: the probability that Selwyn would have a sack marked swag IF he was a robber is, let’s say, 80%. More or less all robbers behave in exactly that way, he argues. The judge, who, while not quite keeping up with the statistician, has also had a few too many, takes this to mean that there is an 80% probability that Selwyn is a robber.

A more sober statistician would recognise the error straight away. The probability that Selwyn would have a sack marked swag IF he was a robber is not the same as the probability that he is a robber IF he has a sack marked swag. There are lots of other reasons why a person might have such a sack: perhaps someone bought it for them as a joke; perhaps they have recently attended a fancy-dress party where they came dressed as a robber; perhaps they had a bit-part in a film… I’m sure you can think of plenty more.

For more on the prosecutor’s fallacy, please check last Sunday’s e-mail.

Sad for Selwyn, the judge is no great fan of joke gifts, fancy-dress parties, or the ‘n’ robbers genre. Selwyn’s last week on Earth is therefore spent in the condemned cell. The judge’s last words were these: ‘Selwyn, I refuse to give you the comfort of knowing which day will be your last. I will tell you this, however: You’ll hang on the day you least expect it!’ (The judge had been reading about the ‘unexpected hanging paradox’.)

Once in his cell, our friend has nothing to do but try to work out which is the day he least expects to be hanged. If today is Monday (it is), the hangman can hardly wait till Sunday. Every other possible day will have elapsed by then and so Sunday can come as no surprise at all. Pleased with his reasoning, Selwyn crosses Sunday off his mental calendar. That leaves Saturday as the last possible day (or, if you prefer, the possible last day). Following the same reasoning, Selwyn dismisses Saturday. It can’t be Saturday, because if the hangman waits until then, once again he’ll have missed every opportunity to surprise his victim. Selwyn therefore crosses off Saturday, too. By exactly this process of reasoning, Selwyn, knowing that the day he least expects to be hanged can never be the last possible day, crosses off all seven days. He relaxes on his cot and falls asleep at last, content in the knowledge that he will not be hanged after all.

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[i] Kahneman, Daniel: Thinking, Fast & Slow, Allen Lane, London, 2011, p250

[ii] Bernstein, Peter L: Against the Gods- The remarkable story of risk, John Wiley & Sons, USA, 1996,p112

[iii] For a really clear explanation, see Mlodinow, Leonard: The Drunkard’s Walk – How randomness rules our lives, Penguin, London, 2008, pp117-120