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?

Sorry, but you have misunderstood the statement of the problem- clearly, as you pointed out, there are trials in which a box containing one match is __not__ selected first by the professor, who instead may choose for say 3 consecutive rounds the other box. __But all those cases are not to be counted__ because the wording of the problem says :

"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? "

meaning that our problem begins if and when the above eventuality has happened, and we are to compute the conditional probability, __given the situation that the professor has first chosen the box with the single match__