I have two different biased coins with probabilities $p_1$ and $p_2$. Coin 1 I toss $n$ times. I would like to know how often I should toss coin 2 to be $p_3$% sure I'll have more heads from coin 2 than from coin 1.
I have read up on binomial distributions and I could figure out the answer by summing those from a guessed starting point and going up or down by trial and error on the computer, but I'm hoping there is an easier way.
Context: Actually the coin flips are archers in a computer game facing off against opposing archers, each having a chance to incapacitate an opponent, I expect there will be anywhere from 1 to 1000 archers on each side. A cautious artificial intelligence wants to know how many archers it has to take so it's reasonably save to go near and have a positive outcome where fewer of his archers fall than those of the enemy.