You will play a coin game against an opponent. A biased coin will be continually flipped where there is a 2/3 chance of Heads and a 1/3 chance of Tails.
If Heads is flipped then you receive \$1 from your opponent. If Tails is flipped then you pay \$1 to your opponent.
You start with \$10 and your opponent starts with \$20. You keep playing until one of you is bankrupt (= has \$0 left); they will be declared the loser, the other will be declared the winner.
What is the probability, that you win?
This seems similar to Gambler's Ruin except the coin is biased. Would this still balance out to 50%? Is there a way to solve this without using stochastic equations?