How do I calculate d20 success probability using the Halfling ‘lucky’ trait with (dis)advantage?

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]: $ $ P + \left(\frac{1}{20}*P\right).$ $ But I’m struggling with deriving $ L_{adv}$ and $ L_{dis}$ . Please can someone show a derivation?