Another Look at the Probabilities of the Notorious Three Prisoners

The "Problem of the Three Prisoners," a counterintuitive puzzle in probability, is reanalyzed, following Shimojo and Ichikawa (1989). Several intuitions that are examined represent attempts to find a simple and commonsensical criterion to predict whether and how the probability of the target event will change as a result of obtaining evidence. However, despite the psychological appeal of these attempts, none proves to be valid in general. A necessary and sufficient condition for change in the probability of the target event, following observation of new data, is proposed. That criterion is an extension to any number of alternatives, of the likelihood principle, that holds in the case of only two complementary alternatives. It is based on comparison of the likelihood of the data, given the truth of the target possibility, with the weighted average of the other likelihoods. This criterion is shown to be psychologically sound, and it may be assimilated to the point of becoming a secondary intuition.