Suppose you have a dice that's biased. Side $1, \ldots 6$ come up with probabilities $p_{1} \ldots p_{6}$ respectively.
I want to model a fair coin. So, I think the best way to do this is to flip twice. If you get something between $1-3$ followed by $4-6$ then it's heads. The other way, it's tails. If you get two things in the same interval, then just reroll twice.
What's the expected number of tosses I'll need to do in terms of $p_{1} \ldots p_{6}$?
Define $P_a = p_1 + p_2 + p_3$ and $P_b = p_4 + p_5 + p_6 = 1 - P_a$. The chance you must re-roll is $P_x \equiv P_a^2 + (1 - P_a)^2$.
The expected number of rolls is:
$${\cal E}(n) = {\sum\limits_{n=0}^\infty n P_x^n \over \sum\limits_{n=0}^\infty P_x^n}$$