A messenger names an excluded candidate while concealing the uniquely selected one. The conditional probability depends on the naming rule. With three equally likely selected candidates and a messenger forbidden to name the observing candidate A, let be the probability of naming B when A is selected. When C is selected the messenger must name B, and when B is selected cannot name B. Thus the probability that A is not selected, conditional on B being named, is . Unbiased naming has and leaves this probability ; bias toward B gives , and bias toward C gives one. Conditioning on the name is different from conditioning only on B's exclusion.
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The Three Prisoners problem is a classic problem in probability and decision theory that illustrates interesting aspects of conditional probability and the paradoxes that can arise in such situations. Here's a typical formulation of the problem: Three prisoners, A, B, and C, are each assigned a number (1, 2, or 3) by a warden, but they do not know their own numbers.