The Inverse Gambler's Fallacy and Cosmology—A Reply to Hacking

Abstract

The gambler's fallacy is committed by one who thinks that a run of one kind of result when rolling dice say makes another outcome more probable. Ian Hacking has argued ('The Inverse Gambler's Fallacy: the Argument from Design. The Anthropic Principle Applied to Wheeler Universes', Mind, I987, pp. 33I-40) that there is also an inverse gambler's fallacy which is committed by one who thinks that an improbable result produced by rolling dice provides good reason for believing that this result comes from the last of a lengthy sequence of rolls. He goes on to argue that this fallacy is involved in a popular objection to the argument from design for God's existence and is also committed by anyone who tries to explain the supposedly improbable character of the universe by reference to John Wheeler's model of sequential universes. Hacking has drawn attention to what is undoubtedly a genuine fallacy, but his claim that this is involved in certain attempts to explain the improbable character of the universe seems to me to be mistaken. The inverse gambler reasons that since an improbable result, such as a double six, is far more likely to occur when a pair of dice is rolled many times than when it is rolled only once, it is reasonable to conclude, when informed that a double six has turned up, that this is the last of a lengthy sequence of rolls. What he overlooks is that every roll of the dice produces an improbable result. That a single roll should produce an improbable result is not itself therefore an improbable happening. In fact it is as certain as anything can be, since no other sort of outcome is possible. An improbable outcome from a single roll of dice requires no further explanation therefore. But does someone who argues that the improbable character of the universe is explained by the fact that it is the last of a lengthy sequence of universes commit this fallacy? I do not think so. Hacking has misrepresented the sort of reasoning employed by someone who appeals to the Wheeler model to explain the delicately balanced nature of the universe we inhabit. This reasoning is not like that of a gambler who says: 'That particular roll of dice produced an improbable result, namely a double six; therefore, it is likely to have been the last of a lengthy sequence of rolls'. Instead it should be compared to the reasoning of someone who at the beginning of a dice-rolling session says to the players: 'I am about to take a nap; please call me if a double six turns up'; and when called, says: 'You must have rolled the dice quite a number of times; otherwise it is unlikely you would have had a double six.' This is perfectly legitimate reasoning. The sleeper knows in advance that he is unlikely to be roused if the dice are rolled only a small number of times. If he has some knowledge of probability theory or is acquainted with the famous exchange between Pascal and Chevalier de Mere, he knows that if the dice are rolled twenty-four times or less, then it is more probable than not that he will remain undisturbed; and if they are rolled twenty-five times or more, then it is more probable than not that he will be