Effects From Extra Die Rolls and Choosing the Highest or Lowest

TL;DR

This paper derives a general formula for the expected value when selecting the highest or lowest die roll outcome over multiple rolls, analyzing how this expectation compares to a single roll as the number of sides increases.

Contribution

It provides a new general formula for the expected value of the maximum or minimum of multiple die rolls with any number of sides, including asymptotic behavior analysis.

Findings

Expected value formulas for maximum and minimum of multiple rolls.

Asymptotic ratio of expected value to number of sides.

Convergence behavior of the ratio as sides increase.

Abstract

This paper looks into the gain or loss from rolling a fair die multiple times and choosing the highest or lowest number as the outcome over rolling the die just once. Specifically, this paper gives a general formula for the expected value of choosing the highest or lowest value of any number of die rolls and sides. It also shows how, for a fixed number of rolls, the ratio between this expected value and the number of sides converges as the number of sides increases asymptotically. The converging behavior of this ratio helps formulate the aforementioned gain or loss.