Player called the number on a d20 before rolling it — was that a 1 in 20 shot or a 1 in 400 shot?

It is 1 in 20, and your friend's arithmetic is correct — he is just answering a different question than the one that happened.

There are indeed 400 equally likely (guess, roll) pairs, each with probability 1/400. But you did not ask for the probability of one specific pair like (17, 17). You asked for the probability of a match, and there are twenty pairs that match: (1,1), (2,2), … (20,20).

So the probability is 20 × 1/400 = 1/20.

The 1 in 400 figure is the answer to "what was the chance he would guess 17 and roll 17, specified in advance". That is a genuine and much rarer event. But nobody specified 17 in advance — the guess was whatever he said, and the interesting thing was that the roll agreed with it.

The intuition that settles it for most people: the guess does not have to be random at all.

Suppose the player always says 17, every single time, forever. Now there is no uncertainty in the guess whatsoever. He is right whenever the die shows 17, which is 1 in 20 rolls.

Making his guess random cannot make him worse at this, and by symmetry it does not make him better either — whatever number he lands on, the die still has a 1 in 20 chance of agreeing. The guess and the roll are independent, so the die does not know or care what was said.

Once you see that a fixed guess gives 1/20, the 1/400 answer becomes clearly wrong, because it would say a player who always guesses 17 is somehow less likely to be right than 1 in 20.

Worth adding the number that will end this permanently: over a session with say 60 rolls, if somebody calls their number each time, the chance of at least one hit is about 95%.

So it is not merely possible, it is close to expected. If it were not happening every few sessions, that would be the surprising outcome.

There is a version of your friend's instinct that is right, and it might be worth conceding at the table so the argument ends properly.

If you want the probability of the specific story — "he will say 17 and then roll 17" — evaluated before he speaks, that genuinely is 1 in 400 (assuming his guesses are uniform). Told that way it sounds far more impressive.

This is the everyday version of a real trap in statistics: choosing which event counts as remarkable after seeing it. The event "he called it" is common. The event "he called 17 specifically" is rare, but it only became a candidate for remarking on because it happened. Twenty different rare events would each have felt equally striking.

So: 1 in 20 for what actually impressed you, 1 in 400 for a prediction nobody made.