538 Dungeons & Dragons Riddler

This problem was the Riddler Classic on 538 for May 15, 2020. The problem is as follows:

The fifth edition of Dungeons & Dragons introduced a system of “advantage and disadvantage.” When you roll a die “with advantage,” you roll the die twice and keep the higher result. Rolling “with disadvantage” is similar, except you keep the lower result instead. The rules further specify that when a player rolls with both advantage and disadvantage, they cancel out, and the player rolls a single die. Yawn!
There are two other, more mathematically interesting ways that advantage and disadvantage could be combined. First, you could have “advantage of disadvantage,” meaning you roll twice with disadvantage and then keep the higher result. Or, you could have “disadvantage of advantage,” meaning you roll twice with advantage and then keep the lower result. With a fair 20-sided die, which situation produces the highest expected roll: advantage of disadvantage, disadvantage of advantage or rolling a single die?
Extra Credit: Instead of maximizing your expected roll, suppose you need to roll N or better with your 20-sided die. For each value of N, is it better to use advantage of disadvantage, disadvantage of advantage or rolling a single die?

This problem seemed like it could be tackled from both a coding/simulation angle and an analytical angle. So I did both. The solutions can be found here; while the path I take is a bit different, the results are the same.

Looking Normal(ly Distributed)

Among all probability distributions, the normal distribution is probably the most well-established and well-characterized. The importance of things like the central limit theorem and the normality assumptions in linear regression highlight it well.

One of the more interesting ones is the fact that you can approximate a binomial distribution with a normal one. Using a continuous distribution to approximate a discrete one feels a little weird, and there are certain assumptions needed for it to work, but it raises an interesting question – how normal can other distributions look?