‘Progress is being made,’ said Ponder.
‘Big amphibians?’ sneered the Senior Wrangler. ‘And things were going so well in the sea. Remember those jellyfish that made nets? And the crabs even had a flourishing land civilization! They had practically got a culture!’
‘They ate captured enemies alive,’ said the Lecturer in Recent Runes, patiently.
‘Well … yes. But with a certain amount of etiquette, at least,’ the Senior Wrangler admitted. ‘And in front of their sand statue of the Great Big Crab. They were obviously attempting to control their world. And what good did it do them? A million tons of white hot ice smack between the eyestalks. It’s so upsetting.’
‘Perhaps they should have eaten more enemies,’ said the Dean.
‘Perhaps sooner or later the planet will get the message,’ said Ridcully.
‘Time for the giant whelks, perhaps?’ said the Lecturer in Recent Runes, hopefully.
‘Big newts is what we’ve got right now,’ said Ridcully. He glanced at the Dean and Senior Wrangler. Ridcully hadn’t maintained his position atop the boiling heap of UU wizardry without a little political savvy. ‘And newts, gentlemen, might be the way to go. Amphibians? At home in the water and on land? The best of both worlds, I fancy.’
The two wizards exchanged sheepish glances.
‘Well … I suppose …’ said the Senior Wrangler.
‘Could be,’ the Dean said grudgingly. ‘Could be.’
‘There we are then,’ said Ridcully happily. ‘The future is newt.’
THIRTY-FOUR
NINE TIMES OUT OF TEN
THERE’S NO NARRATIVIUM on this world.’
Let’s take a step away from the unfolding ancestral tale of The Fish That Came Out From The Sea and look at a more philosophical issue. The wizards are puzzled. On Discworld, things happen because narrative imperative makes them happen. There is no choice about ends, only about means. The Lecturer in Recent Runes is trying to make a sustainable lifeform happen. He thinks that the obstacle to sustainability is the fragility of life – so the only way he can see to achieve this is the two-mile limpet, proof against everything the sky can drop on it.
It never occurs to him that lifeforms might achieve sustainability by other, less direct methods, despite the evidence of his eyes that suggests that a dogged tenacity appears to allow life to arise in the most inhospitable environment, effectively re-creating itself over and over again. The wizards are torn between evidence that a planet is the last place you’d choose to create life, and evidence that life doesn’t agree.
On Discworld, it is clearly recognized that million-to-one chances happen nine times out of ten.1 The reason is that every Discworld character lives out a story, and the demands of the story determine how their lives unfold. If a million to one chance is required to keep that story on track, then that’s what will happen, appalling odds notwithstanding. On Discworld, abstractions generally show up as things, so there is even a thing – narrativium – that ensures that everybody obeys the narrative imperative. Another personification of the abstract, Death, also makes sure that each individual’s story comes to an end exactly when it’s supposed to. Even if a character tries to behave contrary to the story in which they find themselves, narrativium makes sure that the end result is consistent with the story anyway.
What’s puzzling the wizards is that our world isn’t like that …
Or is it?
After all, people live on our world too, and it’s people that drive stories.
As case in point, a story about people who drive. The setting is Jerez Grand Prix circuit, last race of the 1997–98 Formula One motor racing season … Ace driver Michael Schumacher is one Championship point ahead of arch-rival Jacques Villeneuve. Villeneuve’s team-mate Heinz-Harold Frentzen may well play a crucial tactical role. The drivers are competing for ‘pole position’ on the starting grid, which goes to whoever produces the fastest lap in the qualifying sessions. So what happens? Unprecedentedly, Villeneuve, Schumacher, and Frentzen all lap in 1 minute 21.072 seconds, the same time to a thousandth of a second. An amazing coincidence.
Well: ‘coincidence’ it surely was – the lap times coincided. But was it truly amazing?
Questions like this arise in science, too, and they’re important. How significant is a statistical cluster of leukaemia cases near a nuclear installation? Does a strong correlation between lung cancer and having a smoker in the family really indicate that secondary smoking is dangerous? Are sexually abnormal fish a sign of oestrogen-like chemicals in our water supply?
Another case in point. It is said that 84% of the children of Israeli fighter pilots are girls. What is it about the life of a fighter pilot that produces such a predominance of daughters? Could an answer lead to a breakthrough in choosing the sex of your children? Or is it just a statistical freak? It’s not so easy to decide. Gut feelings are worse than useless, because human beings have a rather poor intuition for random events. Many people believe that lottery numbers that have so far been neglected are more likely to come up in future. But the lottery machine has no ‘memory’ – its future is independent of its past. Those coloured plastic balls do not know how often they have come up in previous draws, and they have no tendency to compensate for past imbalances.
Our intuition goes even further astray when it comes to coincidences. You go to the swimming baths, and the guy behind the counter pulls a key at random from a drawer. You arrive in the changing room and are relieved to find that very few lockers are in use … and then it turns out that three people have been given lockers next to yours, and it’s all ‘sorry!’ and banging locker doors together. Or you are in Hawaii, for the only time in your life … and you bump into the Hungarian you worked with at Harvard. Or you’re on honeymoon camping in a remote part of Ireland … and you and your new wife meet your Head of Department and his new wife, walking the other way along an otherwise deserted beach. All of these have happened to Jack.
Why do we find coincidences so striking? Because we expect random events to be evenly distributed, so statistical clumps surprise us. We think that a ‘typical’ lottery draw is something like 5, 14, 27, 36, 39,45, but that 1, 2, 3,19, 20, 21 is far less likely. Actually, these two sets of numbers have exactly the same probability, which for the UK lottery is: 1 in 13,983,816. A typical lottery draw often includes several numbers close together, because sequences of six random numbers between 1 and 49, which is how the UK lottery works, are more likely to be clumpy than not.
How do we know this? Probability theorists tackle such questions using ‘sample spaces’ – their name for what we earlier called a ‘phase space’, a conceptual ‘space’ that organizes all the possibilities. A sample space contains not just the event that concerns us, but all possible alternatives. If we are rolling a die, for instance, then the sample space is 1, 2, 3, 4, 5, 6. For the lottery, the sample space is the set of all sequences of six different numbers between 1 and 49. A numerical value is assigned to each event in the sample space, called its ‘probability’, and this corresponds to how likely that event is to happen. For fair dice each value is equally likely, with a probability of 1/6. Ditto for the lottery, but now with a probability of 1/13,983,816.
We can use a sample space approach to get a ball-park estimate of how amazing the Formula One coincidence was. Top drivers all lap at very nearly the same speed, so the three fastest times can easily fall inside the same tenth-of-a-second period. At intervals of a thousandth of a second, there are one hundred possible lap times for each to ‘choose’ from: this list determines the sample space. The probability of the coincidence turns out to be one chance in ten thousand. Unlikely enough to be striking, but not so unlikely that we ought to feel amazed.
Estimates like this help to explain astounding coincidences reported in newspapers, such as a bridge player getting a ‘perfect hand’ – all thirteen cards in one suit. The number of games of bridge played every week worldwide is huge – so huge that every few weeks the ac
tual events explore a large fraction of the sample space. So occasionally a perfect hand actually does turn up – with the frequency that its small but non-zero probability predicts. The probability of all four players getting a perfect hand at the same time, though, is so micoscopic that even if every planet in the galaxy had a billion inhabitants, all playing bridge every day for a billion years, you wouldn’t expect it to happen.
Nevertheless, every so often the newspapers report a four-way perfect hand. The sensible conclusion is not that a miracle happened, but that something changed the odds. Possibly the players got close to a four-way perfect hand, and the tale grew in the telling, so that when the journalist arrived with a photographer, another kind of narrative imperative ensured that their story fitted what the journalist had been told. Possibly they deliberately cheated to get their names in the papers. Scientists, especially, tend to underestimate the propensity of people to lie. More than one scientist has been fooled into accepting apparent evidence of extrasensory perception or other ‘supernatural’ events, which can actually be traced to deliberate trickery.
Many other apparent coincidences, on close investigation, slither into a grey area in which trickery is strongly suspected, but may never be proved – either because sufficient evidence is unobtainable, or because it’s not worth the trouble. Another way to be fooled about a coincidence is to be unaware of hidden constraints that limit the sample space. That ‘perfect hand’ could perhaps be explained by the way bridge players often shuffle cards for the next deal, which can be summed up as: poorly. If a pack of cards is arranged so that the top four cards consist of one from each suit, and thereafter every fourth card is in the same suit, then you can cut (but not shuffle, admittedly) the pack as many times as you like, and it will deal out a four-way perfect hand. At the end of a game, the cards lie on the table in a fairly ordered manner, not a random one – so it’s not so surprising if they possess a degree of structure after they’ve been picked up.
So even with a mathematically tidy example like bridge, the choice of the ‘right’ sample space is not entirely straightforward. The actual sample space is ‘packs of cards of the kind that bridge players habitually assemble after concluding a game’, not ‘all possible packs of cards’. That changes the odds.
Unfortunately, statisticians tend to work with the ‘obvious’ sample space. For that question about Israeli fighter pilots, for instance, they would naturally take the sample space to be all children of Israeli fighter pilots. But that might well be the wrong choice, as the next tale illustrates.
According to Scandinavian folklore, King Olaf of Norway was in dispute with the King of Sweden about ownership of an island, and they agreed to throw dice for it: two dice, highest total wins. The Swedish king threw a double-six. ‘You may as well give up now,’ he declared in triumph. Undeterred, Olaf threw the dice … One turned up six … the other split in half, one face showing a six and the other a one. ‘Thirteen, I win,’ said Olaf.2
Something similar occurs in The Colour of Magic, where several gods are playing dice to decide certain events on the Discworld:
The Lady nodded slightly. She picked up the dice-cup and held it steady as a rock, yet all the Gods could hear the three cubes rattling about inside. And then she sent them bouncing across the table.
A six. A three. A five.
Something was happening to the five, however. Battered by the chance collision of several billion molecules, the die flipped onto a point, spun gently and came down a seven.
Blind Io picked up the cube and counted the sides.
‘Come on,’ he said wearily. ‘Play fair.’
Nature’s sample space is often bigger than a conventional statistician would expect. Sample spaces are a human way to model reality: they do not capture all of it. And when it comes to estimating significance, a different choice of sample space can completely change our estimates of probabilities. The reason for this is an extremely important factor – ‘selective reporting’, which is a type of narrativium in action. This factor tends to be ignored in most conventional statistics. That perfect hand at bridge, for instance, is far more likely to make it to the local or even national press than an imperfect one. How often do you see the headline BRIDGE PLAYER GETS ENTIRELY ORDINARY HAND, for instance? The human brain is an irrepressible pattern-seeking device, and it seizes on certain events that it considers significant, whether or not they really are. In so doing, it ignores all the ‘neighbouring’ events that would help it judge how likely or unlikely the perceived coincidence actually is.
Selective reporting affects the significance of those Formula One times. If it hadn’t been them, maybe the tennis scores in the US Open would have contained some unusual pattern, or the football results, or the golf … Any one of those would have been reported, too – but none of the failed coincidences, the ones that didn’t happen, would have hit the headlines. FORMULA ONE DRIVERS RECORD DIFFERENT LAP TIMES … If we include just ten major sporting events in our list of would-be’s that weren’t, that one in ten thousand chance comes down to only one in a thousand.
Having understood this, let’s go back to the Israeli fighter pilots. Conventional statistics would set up the obvious sample space, assign probabilities to boy and girl children, and calculate the chance of getting 84% girls in a purely random trial. If this were less than one in a hundred, say, then the data would be declared ‘significant at the 99% level’. But this analysis ignores selective reporting. Why did we look at the sexes of Israeli fighter pilots’ children in the first place? Because our attention had already been drawn to a clump. If instead the clump had been the heights of the children of Israeli aircraft manufacturers, or the musical abilities of the wives of Israeli air traffic controllers, then our clump-seeking brains would again have drawn the fact to our attention. So our computation of the significance level tacitly excludes many other factors that didn’t clump – making it fallacious.
The human brain filters vast quantities of data, seeking things that appear unusual, and only then does it send out a conscious signal: Wow! Look at that! The wider we cast our pattern-seeking net, the more likely it is to catch a clump. For this reason, it’s illegitimate to include the data that brought the clump to our attention as part of the evidence that the same clump is unusual. It would be like sorting through a pack of cards until you found the ace of spades, putting it on the table, and then claiming miraculous powers that unerringly accomplish a feat whose probability is one in 52.
Exactly this error was made in early experiments on extra-sensory perception. Thousands of subjects were asked to guess cards from a special pack of five symbols. Anyone whose success rate was above average was invited back, while the others were sent home. After this had gone on for several weeks, the survivors all had an amazing record of success! Then these ‘good guessers’ were tested some more. Strangely, as time went on, their success rate slowly dropped back towards the average, as if their powers were ‘running down’. Actually, that effect wasn’t strange at all. It happened because the initial high scores were included in the running total. If they had been omitted, then the scoring rate would have dropped, immediately, to near average.
So it is with the fighter pilots. The curious figures that drew researchers’ attention to these particular effects may well have been the result of selective reporting, or selective attention. If so, then we can make a simple prediction: ‘From now on, the figures will revert to fifty-fifty.’ If this prediction fails, and if the results instead confirm the bias that revealed the clump, then the new data can be considered significant, and a significance level can sensibly be assigned by the usual methods. But the smart money is on a fifty-fifty split.
The alleged decline in the human sperm count may be an example of selective reporting. The story, widely repeated in the press, is that over the past 50 years the human sperm count for ‘normal’ men has halved. We don’t mean selective reporting by the people who published the first evidence – they took pains to avoid all
the sources of bias that they could think of. The ‘selective reporting’ was done by researchers who had contrary evidence but didn’t publish it because they thought it must be wrong, by journal referees who accepted papers that confirmed a decline more often than they accepted those that didn’t, and by the press – who strung together a whole pile of sex-related defects in various parts of the animal kingdom into a single seamless story, unaware that each individual instance has an entirely reasonable explanation that has nothing to do with falling sperm-counts and often nothing to do with sex.
Sexual abnormalities in fish near sewer outlets, for instance, are probably due to excess nitrites, which all fish-breeders know cause abnormalities of all kinds – and not to oestrogen-like compounds in the water, which would bolster the ‘sperm count’ story. Current data from fertility clinics, by the way, show no signs of a decline.
Humans add narrativium to their world. They insist in interpreting the universe as if it’s telling a story. This leads them to focus on facts that fit the story, while ignoring those that don’t. But we mustn’t let the coincidence, the clump, choose the sample space – when we do that, we’re ignoring the surrounding space of near-coincidences.
Jack and Ian managed to test this theory on a trip to Sweden. On the plane, Jack predicted that a coincidence would happen at Stockholm airport – for reasons of selective reporting. If they looked hard enough, they’d find one. They got to the bus stop outside the terminal, and no coincidences had occurred. But they couldn’t find the right bus, so Jack went back to the enquiries desk. As he waited, someone came up next to him – Stefano, a mathematician who normally occupied the office next door to Jack’s. Prediction confirmed. But what was really needed was evidence of a near-coincidence – one that hadn’t happened, but could have been selectively reported if it had. For instance, if some other acquaintance had shown up at exactly the same time, but on the wrong day, or at the wrong airport, they’d never have noticed. Near coincidences, by definition, are hard to observe … but not impossible. Ian happened to mention all of the above to his friend Ted, who was visiting soon after. ‘Stockholm?’ said Ted: ‘When?’ Ian told him. ‘Which hotel?’ Ian told him ‘Funny, I was staying there one day later than you!’ Had the trip been one day later, the ‘coincidental’ encounter with Stefano wouldn’t have happened – but the one with Ted would.
The Science of Discworld Revised Edition Page 29