ICOTS-7, 2006: Loschi, Iglesias, and Wechsler 1 UNPREDICTABILITY, PROBABILITY UPDATING AND THE THREE PRISONERS PARADOX Rosangela H. Loschi Universidade Federal de Minas Gerais, Brazil Pilar L. Iglesias Pontificia Universidad Católica, Chile Sergio Wechsler Universidade de São Paulo, Brazil loschi@est.ufmg.br This paper discusses the Three Prisoners paradox in the light of three different procedures for the updating of probabilities - Bayesian conditioning, superconditioning and Jeffrey's rule - as well as assuming the unpredictability of receipt of information by prisoner A. The formulation of the paradox in this temporal setting brings new insight to the problem and, on the other hand, the paradox is a good way to explain the different updating probability procedures and the difference between conditional probabilities and posterior distributions. INTRODUCTION The three prisoners. Two are to be shot and the other freed; none is to know his fate until the morning. Prisoner A asks the warden to confide the name of one other than himself who will be shot, explaining that as there must be at least one, the warden won’t be giving away anything relevant to A’s own case. The warden agrees, and tells him that B will be shot. This cheers A up a little, by making his judgement probability for being freed rise from 1/3 to 1/2. But that’s silly: A knew already that one of the others would be shot, and (as he told the warden) he’s no wiser about his own fate for knowing the name of some other victim.” (Jeffrey, 1992, p. 122) The Three Prisoners paradox, just presented, is an old problem from Probability Calculus which has been discussed from many different points-of-view. This “paradox” is also an excellent tool in teaching some different procedures for the updating of probabilities and the difference between posterior distributions and conditional probabilities. On the other hand, these updating procedures provide new insight in the paradox. Apparently the answer provided by prisoner A is a contradiction since, as prisoner A tells the warden, the information given about the other two prisoners does not apprise prisoner A of his own condition. Thus, the prisoner A’s opinion about the event “A will live” ought to be the same after the receipt of the information provided by the warden, i.e., the posterior probability of this event should also be 1/3. On the other hand, we should notice that the solution presented by prisoner A does not change the prior opinion of indifference among the prisoners revealed by the prisoner A’s prior distribution. What is the right answer? According to Jeffrey (1992), prisoner A follows erroneously the council of parochialism - that is, prisoner A constructs his posterior distribution using Bayesian conditioning. Prisoner A considers the sample space 1 Ω = {A, B, C}, where X represents “X will live,” X = A, B, C. On this space the information provided by the warden does not generate a sufficient partition for {P, P*}, which makes the use of Bayesian conditioning inappropriate. (See de Finetti (1972), for the difference between Bayes’ formula and Bayesian Conditioning.) Some alternative procedures for the updating of probabilities are proposed in the literature. See, for example, Diaconis and Zabell (1982), Jeffrey (1992), Howson and Urback (1993) and others. However, there is no guidance for coherent temporal behavior which produces an inevitable probability updating procedure (Goldstein, 1985; Dawid, 1985). Consequently, prisoner A may update his/her prior opinion by means of a complete reassessment of his/her opinion. This paper extends previous works by presenting alternative explanations to the Three Prisoners paradox using Bayesian conditioning, superconditioning and Jeffrey’s rule as well as by