Here is a comprehensive DPR calculator, and here is the mathematics behind it. I’m trying to follow along with the equations.

At the bottom of the second page are formulas for success probability $L$ of a Halfling (who has luck) in normal circumstances and with advantage and disadvantage: $$L = P + frac{1}{20}P,$$ $$L_{adv} = P_{adv} + left(frac{2}{20}(1 – P) – frac{1}{400}right)P,$$ $$L_{dis} = P_{dis} + frac{2}{20}P^2,$$ where:

- $P$ is the probability of succeeding on any single roll,
- $P_{adv} = 1 – (1 – P)^2$ is the probability of succeeding with advantage (not failing both rolls), and
- $P_{dis} = P^2$ is the probability of succeeding with disadvantage (succeeding both rolls).

The $P$s are quite easy to derive, and $L$ is just *passing outright* OR (*rolling a 1* AND THEN *passing the reroll*): $$L = P + left(frac{1}{20}*Pright).$$ But I’m struggling with deriving $L_{adv}$ and $L_{dis}$. Please can someone show a derivation?