The Science of Discworld Revised Edition

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 actual 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.