Expected Number of Dice Rolls Until an Increasing Run of Three

Daniel Chen

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arXiv · 2308.14635

Expected Number of Dice Rolls Until an Increasing Run of Three

Daniel Chen

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Abstract

A closed form is found for the expected number of rolls of a fair n-sided die until three consecutive increasing values are seen. The answer is rational, and the greatest common divisor of the numerator and denominator is given in terms of n. As n goes to infinity, the probability generating function is found for the limiting case, which is also the exponential generating function for permutations ending in a double rise and without other double rises. Thus exact values are found for the limiting expectation and variance, which are approximately 7.92437 and 27.98133 respectively.

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Daniel Chen. 2023-10-18. Expected Number of Dice Rolls Until an Increasing Run of Three. https://arxiv.org/abs/2308.14635

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