How many times must a 6-sided die be rolled until 1 side is expected to appear at least twice and 2 side is expected to appear at least once?
We know the simple question for 1 and 2 side appear at least once is $\frac{1}{1/6}+\frac{1}{1/6}-\frac{1}{1/6+1/6}=9$
the appearance probability for 1 and 2 side is $p_a, p_b$, someone gives a general formula for primal problem:$\frac{2}{p_a}+\frac{1}{p_b}-\frac{1}{p_a+p_b}-\frac{p_a}{p_a+p_b}·\frac{1}{p_a+p_b}=12+6-3-1.5=13.5$ (it's correct when you use vba to test)
so how to understand the part of $\frac{p_a}{p_a+p_b}·\frac{1}{p_a+p_b}$?