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The pills problem revisited. (English) Zbl 1054.05004

In the pills problem, as stated by Knuth and McCarthy [Am. Math. Mon. 98, 264 (1991)], a bottle contains \(m\) large pills and \(n\) small pills, where a large pill is equivalent to two small pills. Each day a pill is chosen at random. If a small pill is chosen it is eaten; if a large pill is chosen, it is broken into two halves, one of which is eaten and the other half , now considered as a small pill, is returned to the bottle. What is the expected number of small pills in the bottle when no large pills remain? The solution was found by Hesterberg, Stromquist and Wagner [Am. Math. Mon. 99, 684 (1992)]. The present paper shows how to find higher moments as well as the expected value, and goes on to discuss generalisations of the problem, to the case where each large pill is equivalent to \(p\) small pills, and to the case where there are 3 sizes of pill. Recursion methods are used.

MSC:

05A15 Exact enumeration problems, generating functions
60C05 Combinatorial probability
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