There are numerous things which are unclear.

In my experience of discussions of the “unexpected hanging” paradox, there is a prevalent assumption that either the judge or the condemned man is wrong, but not both; the problem is presented as if the question is “which is wrong?”. I think both are wrong.

In particular, I think the judge is wrong to claim that the condemned man will be surprised by the executioner knocking on his cell door at noon on the day of the execution. How does he know? Given the paradoxical evidence, what makes the judge so confident of how exactly the condemned man will resolve the paradox, if at all?

What constitutes evidence that the judge was wrong? By the wording of the question, if the executioner turns up but this does not surprise the condemned man, then that is evidence. So, suppose that on each weekday, just before noon, the condemned man says “Good morning, Mr. Executioner. I’ve been expecting you.”? On each day when the executioner is not there, does that make the condemned man wrong? But on the day of the hanging, does that make the judge wrong?

In statements of the paradox that I have read, where someone narrates the story of what happens after the judge’s decree, the narrator says that, one day, the executioner turns up, to the condemned man’s surprise. The narrator then implies that the judge’s decree is seen to be correct. This implication is invalid. That decree is tantamount to the general claim that, in every possible scenario, the executioner’s arrival would surprise the condemned man. Even if the executioner’s arrival turns out to surprise the condemned man in the real scenario, that is just one particular instance, & it does not prove the judge’s general claim.

When the judge speaks to the condemned man, is the date of the execution set, & does the judge tell him this? If not, then the condemned man is wrong to argue “Suppose my execution is set for Friday. Then…” and similarly for the other days.

The paradox can be recast as the following game (it’s easier to play & doesn’t entail any inconvenient corpse-disposal). One player (the judge) takes 5 cards, one of which represents the hanging (for added realism, use a tarot pack & make one of them Le Pendu). As the previous paragraph suggests, there are actually two variants. In one, this player shuffles them; in the other, the player secretly looks at them. In each case he deals them face down in a row. Then, for each card in turn, he asks the other player (the condemned man) whether he wants to choose it. Then the judge turns that card face up.

• If it’s the “hanging” card, then if the condemned man chose it, he wins (for proving that the judge was wrong to say that the hanging would surprise him) but if not, the judge wins.
• Otherwise, if the condemned man chose it, he loses for having made a wrong choice, but if not, the judge proceeds to the next card, if any.

What’s the probability of the condemned man winning?

I make it 1/5 in both variants. Judge and condemned man were both a bit wrong, but the judge was less wrong than the condemned man.