Terence Tao's comment is very much worth reading. He addresses "is wrong". (It's not that simple.) Our intuition about probability fails us because it involves

Terence Tao's comment is very much worth reading. He addresses "is wrong". (It's not that simple.) Our intuition about probability fails us because it involves non-measurable sets, for which probabilities can't be properly assigned. Then, the intuition that "each prisoner has a 50% chance of being right" (based on a "rigid motion" argument in (Z_2)^N, or the space of sequences of {0, 1}) fails us because we don't have a meaningful probability. (We're conditioning on an equivalence class that is similar to a Vitali set; it can neither be assigned a zero nor positive measure, so it's non-measurable.)