Simple coinflipping problem.

[Mr_J]

Say I decide to enter a coinflipping contest. We both start with $15. Say we are forced to wager $1 on every coinflip. I'm using a rigged coin, so I will win 55% of these coinflips on average. We flip coins until one of us loses the $15. How often will I take his money (do the math in the post)???

How much does my winrate change if we were forced to wager just 50 cents?