Professor X smokes a pipe. He carries two identical matchboxes, originally containing 20 matches each. When he lights his pipe, he chooses a matchbox at random and lights his pipe with one match and discards the used match.

There will eventually arise an occasion when he first selects a matchbox with only one match in it. At this point, what is the expected number of matches in the other box?

This is a variant of the Banach Matchbox Problem. One needs to decide what the problem here is requesting: does "expected number of matches" mean the number of matches with the highest probability? Otherwise, we would presumably want a frequency distribution of number of matches from 1 to N in the "other" box, assuming he has been counting the number of trials -- but if he had been doing that, he would already know the number left in the other box (trials, less 19 selections of the box left with one).