Is there a 1 in 20 or 1 in 400 chance of guessing the outcome of a d20 roll before it happens? My friends are in a bit of an argument over Dungeons & Dragons.
My player managed to guess the outcome of a D20 roll before it happened, and my friend said that his chance of guessing the number was 1 in 20. Another friend argues that his chance of guessing the roll is 1 in 400 because the probability of him randomly guessing a number and then rolling it were both 1 in 20 so the compound probability is 1 in 400. Which of these probabilities is a better characterization of our situation, and what were really his chances?

The probability of your friend correctly predicting the outcome of the roll was 1 in 20.
The probability of your friend correctly predicting that that specific number would be the outcome of the roll was 1 in 400.
Let's say the friend predicted it would be a 20. If you'd have been equally impressed if he'd predicted it would be a 4, then you're in the first situation (1/20). If you don't care about a correct prediction unless it was a 20, you're in the second situation (1/400).
Put another way, suppose you were watching your friend from behind a glass window, and you knew he was going to try to call the roll. If you said, "he will correctly predict the outcome of the roll" you'd have a 1 in 20 chance of being right. If you said, "he will correctly predict that he will roll a perfect 20" you'd have a 1 in 400 chance of being right.
