Effects From Extra Die Rolls and Choosing the Highest or Lowest

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.