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expected value of the number of small pills in the bottle

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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 for small pills and another part gets eaten.

What is the expected value of the number of small pills in the bottle after the last large pill has been chosen and its smaller half returned??

And if Y is the day on which the last large pill is chosen. What is E[Y]?

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The solution shows how to calculate the expected value of the number of small pills in the bottle after the last large pill has been chosen and its smaller half returned.

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