Abstract:
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In this paper, we will deal with three seemingly different problems: the guessing problem, which involves correctly guessing an integer chosen from the set of integers {1,2,…,n}; the collector’s problem, which requires computing the time needed to collect all different n items in a collection; and the selection problem, which entails choosing a ball from a collection of balls of two different colors in a jar.
Although these problems seem to be totally unrelated, they do, in fact, exhibit one common attribute: their expected values are computed in terms of harmonic numbers.
We start our paper by a brief mathematical prelude where we discuss the basic properties of harmonic numbers and the linearity of expectance. Following that, we compute the expected values for the aforementioned problems. Finally, we explore some pedagogical implications of these outcomes.
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