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).  

Posted by ed bottemiller on 2010-03-30 14:00:20