Monday, November 22, 2010

jacob bernoulli on luck

The mathematics of probability began in the 1650s with the correspondence between Fermat and Pascal on the problem of points. The crucial question, first proposed to Pascal by the Chevalier de Méré, concerned the question of how to divide the pot if a gambling game had to be interrupted before it could be completed, e.g. suppose $x is the pot, 7 rounds are needed to win, player A has won 6 rounds, while player B has won 5 rounds: if they are forced to interrupt their game at this stage, what constitutes a fair division of the pot?

All early treatises (and even most modern ones) addressed the problem of points in their presentation of the probability calculus. For example, Christiaan Huygens places a discussion of the problem of points near the very start of one of the first treatises on probability, his De ratiociniis in ludo aleae (1657).

In 1713, Jacob Bernoulli's treatise, Ars Conjectandi was published posthumously. This was the first work to prove a limit theorem about probability, as well as extending probability theory to cases where probabilities where unequal, but could be derived from underlying equipossibility. The first of Ars Conjectandi's four section reprints Huygens' treatise with extensive commentary and solutions to all problems. When Huygens presents the problem of points, he points out that only the remaining games need be considered (and not the number of games already played), a remark which Huygens supplements with the comment:
In general, we should take no account of past games when we compute the lots for games that are all in the future. For in any new game the probability that fortune will continue to favor those that it has favored before is no great than the probability that it will favor those who have been the most unfortunate. I observe this in opposition to the ridiculous opinion of the many who think of fortune as some kind of habit, which remains in a person for a while and somehow gives him almost a right to expect similar fortune to continue.

Of course, Bernoulli is absolutely right, a point which can even be empirically determined, e.g. as in the hot hand fallacy.

An interesting question, though, is the balance of opinion on the question of streaks of luck vs. the gambler's fallacy, which expects streaks to reverse, or be balanced by future outcomes. Of course, from both the mathematical and the empirical standpoints, probabilities are not affected by streaks or runs in the outcomes. But, from a sociological standpoint, is belief in luck (good or bad) more common than belief in the gambler's fallacy? Given that they are mutually contradictory, how frequently can both beliefs be found in the same individual? If belief in one statistically dominates belief in the other, why? Underlying optimism (pessimism)? If our beliefs about such matters are derived from our experience with the world, why the tendency to distort rather than simply probability match (especially given that humans can be quite good at probability matching under other circumstances)?

No comments: