A bottle initially contains m large pills and n small pills. Each day, a patient randomly chooses one of the pills. If a small pill is chosen, then that pill is eaten. If a large pill is chosen, then the pill is broken in two; one part is returned to the bottle (and is now considered a small pill) and the other part is then eaten.
(a) Let X denote the number of small pills in the bottle after the last large pill has been chosen and its smaller half returned. Find E[X].
Define n + m indicator variables, one for each of the small pills initially present and one for each of the m small pills created when a large one is split in two. Now use the argument of Example 2m.
(b) Let Y denote the day on which the last large pill is chosen. Find E[Y].
What is the relationship between X and Y?