31 The Gamblers Mistake

"If the chances of a coin coming tails are exactly one in two, and we have just had a run of nine straight heads, what are the chances of heads coming up the next throw?" "Let me have a pencil." "You don't need to calculate. The chances are one in two." "You mean that it is one in two to have ten straight heads?" "No, the chances of that are less than one in a thousand,"

There is no contradiction here. The gambler who bets his fortune that the next throw will be tails has an even chance of losing it. The series of heads has no effect at all on the next throw. If in advance he had bet against ten consecutive throws, he probably would have collected his bet; it would be a sound investment to give odds of several hundred to one. This fallacy has been discussed ever since the invention of roulette and the consequent discovery of probability theory by Pascal. But perhaps the little dialogue above brings out all that is important to remark here.

There is a variation of the gambler's fallacy consisting of the failure to realize that a slight possibility is, nevertheless, still a possibility. This is a sort of failure of imagination. Of course, a man directs his main efforts to meeting those conditions which seem most probable, but, in important matters, he should not dismiss the improbable. The unlikely may be what actually happens.

EXAMPLECOMMENT
When the presence of unidentified aircraft in the vicinity of the Hawaiian Islands was indicated by radar on the morning of December 7, 1941, the officer in charge disregarded the evidence on the grounds that hostile planes could not be anywhere near. Great generals have won battles by outwitting less imaginative opponents whose preparations were limited to taking care of only what was likely to happen. Washington's crossing the Delaware to capture the Hessians is an instance, from American history, of taking advantage of this. History affords hundreds of similar examples. Poor generals lapse into this fallacy; great ones use it by consciously outimagining their opponents.
Peter excuses his marrying again after his fifth divorce by saying, "Statistics show that at least half of American marriages are happy. My luck is bound to change this time." Even if the chances were nine out of ten, he would be bound to do nothing at all. He might still have bad "luck" in his choice of brides. As a matter of fact, if the statistics Peter quotes are correct, he still has about an even chance of needing another divorce. A likely explanation for Peter's persistent bad luck is a character flaw that bars him from achieving a happy home life -- no matter what the general statistics reveal. Unlike the tossed coin which has no ties with the past, Peter's matrimonial problems surely have.