(Hint: the chance doesn't have to be 50:50.)
Make your decision as follows: if in two flips the coin comes up heads first and then tails, go with person A. If it comes up tails first and then heads, go with person B. If it comes up both heads or both tails, ignore the flips and try again.
The probability of going with person A is now the probability that heads comes up times the probability that tails comes up, so that’s
The probability of going with person B is the probability that tails comes up times the probability that heads comes up, which is of course the same as the probability of going with person A:
You might have to flip for a while, since you have to ignore the heads-heads and tails-tails outcomes, but at least you can be sure that A and B have the same chance of being picked.
The same works for any pair of probabilities for heads and for tails. In each case the chance of each person being picked is