The game of red and blue revisited: propensities, subjective probabilities, and the goodness-of-fit paradox

“The game of red and blue” is an example given by Gillies (2000) to illustrate the superiority of his propensity interpretation of probability over the subjective interpretation of de Finetti. I will show that the falsifiability of probability models by means of statistical tests, which is crucial for Gillies’ interpretation, is subject to serious difficulties, one of which is the goodnessof-fit paradox: The confirmation by a statistical (goodness-of-fit) test refutes the validity of a probability model. I conclude that the existence of objective probabilities is an irreducibly metaphysical assumption, i.e., it is strictly not testable by observations. On the other hand, I agree with Gillies that the example is a valid illustration of a shortcoming of the subjective interpretation, namely that the requirement that every change of a probability assignment has to follow Bayes’ rule is too inflexible if data is observed that is inconsistent with the prior distribution. Instead of rejecting the two discussed interpretations of probability, I argue that probability modeling inevitably requires subjective decisions about accepting such untestable assumptions. The value of the models is their power to structure our thoughts and our perceptions of the uncertain, and not their correspondence with any objective reality.