Each round that was won by a biased coin flip where $1 would go to the winner. The coin flip was biased 2/3 towards me. What's the probability that you win the game by winning all of the money?

1/3

Step-by-step explanation:

The total probability of getting either a head or a tail after flipping a coin = 1

And given that 2/3 has already been biased towards a person. The probability that the other wins the game would simply be

1 - 2/3 = 1/3. hence allowing the other person win the $1.