Afterthought
By ALAN BRIEN
I NEVER cease to be sur- prised at the limitations of common sense. Beneath the rational, reasonable, sceptical world that we the tolerably intelligent and
well-informed inhabit, where miracles never hap- pen, coincidences have no significance, dreams do not foretell the future, and no one can be in two places at once, there lies an irrational, improbable under- world of faith serviced by physicists and mathe- maticians. Here a perspex curtain separates the two cultures. On top, we assume that doubt can move mountains. Like Dr. Johnson, we kick the solid stone and growl, 'Thus I refute you,' at solipsists, spiritualists, mystics and idealist philo- sophers. Down below, in topsy-turvy land, tunnelling underneath our sandy feet. scientists mine away discovering three impossible things every day before breakfast and unearthing anti- matter, mirror-universes where left does not re- flect right, implosions where time tuns backwards, and laws of probability which seem to defy every- day assumptions.
Much of what is brought to the surface by these brainy moles I frankly do not understand —even when it is translated into simple-minded, milk-bar parables by the scientific correspondents of Sunday newspapers. But I think all of us share the surprise expressed by my old friend, former colleague and fellow ignoramus, Peter Chambers, in his column in Monday's Daily Express when he described how he lost £1 betting that out of sixteen people in a bar no two would have the same birthday. Such was his confidence that the odds must be in his favour that he doubled his stake on a second sixteen drinkers—and lost again. On cross-referencing the thirty-two, he was fur- ther mystified to learn that there were, in fact, three sets of identical birthdays.
As it happens, my mother-in-law had already warned me against this particular trap. And I have since investigated the reasoning which justifies Mr. Chambers's cunning pal in his money-making lure. It can be found in two useful paperbacks now on sale—The New World of Mathematics, by George A. W. Boehm (Faber, 5s.), and Use and Abuse of Statistics, by W. J. Reichmann (Pelican 5s.). From these sources I suggest that anyone who seeks to repeat the trick, with a better than evens chance of taking a sucker to the cleaners, should insist on a rather larger group. Mr. Reichmann's argument is the clearer of the two and I follow him. He starts with a crowd of thirty and proposes that, with 365 dates available, most of us would assume the chance of duplication as about one in twelve.
However, moreover, and be that as it may, he explains that the easiest way to calculate the ratio is by working out the probability of total failure. The first man has 365 birthdays to choose from. The second has 364 available without over- lapping. And so on. Reducing by one at each remove, the thirtieth man has 336 chances in 365 of not sharing a birthday with the preceding twenty-nine men. The total probability is there- fore the product of all twenty-nine terms: 364 363 362 336
—x—x—x...x-
365 365 365 365 This is equivalent to 0.3 for the probability of failure. The probability of success must be 0.7 --in other figures, there are approximately seven chances out of ten that at least two birthdays will be shared among thirty people.
This is all very well. Mr. Chambers has demonstrated that this forecast of probability operates in practice. Mr. Boehm and Mr. North and many other mathematicians have provided a chain of argument which proves that there is no magic in the result. Nevertheless, and not- withstanding, and be that as it may, the whole thing still smacks to me of jiggery-pokery. Before I would risk more than half a crown on the same bet I would need 183 people to feel a safe winner. I know I am wrong. But like Mr. Chambers, I do not feel wrong.
The same ordinary man's common sense for- bids me to accept that, no matter how many heads have turned up in a row, the chance of the next penny coming down tails is still evens. Our statistician admits thab the probability of tossing, say three heads in a row'can be worked out mathematically. For the first, it will be 1; for the second, + X -1 =-4-; for the third, *xixf
But, he says, if-we stop after the second throw, though the chances of a hat-trick remain *, the chance of the third head on its own is now —eveg,,though this would complete the sequence. It appears that the odds in favour of heads are increasing, not diminishing. The statistician replies that probability does not have meaning in the past. The first two throws have been facts— and the value of a certainty is 1 just, as the value of an impossibility is 0. So the probability is IXIxf =4. But to us pragmatic men (whose whole life, from crossing the street to crossing the boss, depends on a successful estimate of adds in our favour) it seems that two results are only equi-probable if there is a definite reason for thinking them so. The evidence for heads-or- tails probability for us must lie in observation. Tossing experiments have been carried out: in a typical sequence of 8,000 pennies, two runs each of fifteen consecutive tails were counted. We would then, in our rough-and-ready way, estimate the probability of twenty consecutive tails as near zero. Supposing our opponent had tossed nineteen, would we not be fools to avoid betting 10-1 on heads next time?
And supposing he had tossed 1,000 tails, how would we calculate the probability on the 1,001st? Can we still hold with Mr. Reichmann that 'a penny is free to react in different ways right up to the action of being tossed'? On the one hand, it could be argued that this coin had shown an improper desire and bias to land head down, therefore the odds in favour of it continuing to do so were very high. Or it could be claimed that this mutiny against probability must soon be suppressed by the ghost of Pascal and the chances of a head were mounting astronomically. Either way, evens would surely be the least probable of forecasts. Aristotle said the probable was what usually happened. When it ceases to happen—as in a long run of heads or taffs—the canny man revises his expectations of what will occur afterwards. The difference between us and the statisticians is that we are concerned about the probability of events, they are concerned about the probability of pro- positions about events. What matters to us is truth, what matters to them is consistency. It is
the difference between Bertrand Russell as a moralist and Bertrand Russell as a mathema- tician. Did he not define mathematics as 'the subject in which we never know what we are talking about, nor whether what we are saying is true'? I fancy he would not accept so happily the same definition of his politics.
Previous page
