A paradox of confirmation

Abstract

Two out of three prisoners, a, b and c, are to be exiled, while the third goes free. The choice of the two has been determined by a fair draw, of whose outcome they are ignorant. Prisoner a is keen to find out his fate, and attempts to question the guard, who knows the result of the draw. Knowing that at least one of the other two prisoners will be exiled, a asks the guard to tell him a name, either b or c, of a person who will be exiled. The guard, who is forbidden to converse with the prisoners, refuses to be drawn out. However, a, who is an amateur probability theorist, reasons to himself as follows: "If the guard had named b, I would have concluded that my chance of being freed was 1/2. If the guard had named c, I would have reasoned analogously. Either way, my likelihood of being freed is 1/2." How has this thought-experiment raised a's probability of freedom from 1/3 to 1/2?1 When faced with such a paradox, one possible strategy is to embrace its conclusion, and endorse a's reasoning. We would then have to expose the error in the initial distribution of probabilities, drawing, perhaps, on some subtle feature of the story. However, the existence of the guard is merely a narrative embellishment. Indeed, we can imagine the prisoner reasoning about a hypothetical guard, and it is hard to see how the a priori distribution of equal probabilities to each prisoner's going free given the knowledge that their fate is to be decided by a fair draw can be faulted. The guard, whether hypotheti? cal or real, can make no difference. Worse still, analogous reasoning can be used to undermine the very possibility of assigning probabilities. To see this, we begin by suppos