We revisit the pills problem proposed by Knuth and McCarthy. In a bottle there are m large pills and n small pills. The large pill is equivalent to two small pills. Every day a person chooses a pill at random. If a small pill is chosen, it is eaten up, if a large pill is chosen it is broken into two halves, one half is eaten and the other half which is now considered to be a small pill is returned to the bottle. How many pills are left, on average, when the last large pill has disappeared? We show how to compute the moments, in particular the variance, and then generalize the problem in various ways.

The pills problem, proposed by Knuth and McCarthy, involves a bottle containing m large pills and n small pills, with a person randomly choosing a pill each day, and the remaining pills are discarded as small pills or broken into two halves, resulting in an average of n small pills left after the

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The Pills Problem Revisited

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