How do I calculate d20 success probability using the Halfling 'lucky' trait with advantage?

Question

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. With advantage it is $$L_{adv} = P_{adv} + \left(\frac{2}{20}(1-P) - \frac{1}{400}\right)P,$$ where:

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

In my attempt for deriving this (below), I have a sign error. Please can someone explain where I've gone wrong/show a correct derivation?

To succeed you require:

• succeeding outright while advantaged, OR
• rolling a \$1\$ with die \$a\$ AND failing with die \$b\$, AND THEN succeeding the reroll, OR
• rolling a \$1\$ with die \$b\$ AND failing with die \$a\$, AND THEN succeeding the reroll, OR
• rolling two \$1\$s AND THEN succeeding a reroll: $$L_{adv} = P_{adv} + \frac{1}{20}*(1-P)*P + \frac{1}{20}*(1-P)*P + \frac{1}{400}*P\\= P_{adv} + \left(\frac{2}{20}(1-P) + \frac{1}{400}\right)P$$

Answer 1

The case where you roll two 1s is actually included in both of the previous two cases (rolling 1 on a, failing on b; rolling 1 on b, failing on a).

Consequently, it has been counted twice and needs to be subtracted from the total instead of added separately.

More generally, this is an example of the Inclusion-Exclusion Principle.