6 Arrows

You and your friend are competing in an archery competition. You are a more skilled archer than he is, and the distances of your arrows to the center of the bullseye are i.i.d. (independent and identically distributed) Uniform[0, 1] whereas his are i.i.d. Uniform[0, 2]. To even out the playing field, you both agree that you will shoot one arrow and he will shoot two. The arrow closest to the center of the bullseye wins the competition. What is the probability that you will win? Note: The distances from the center of the bullseye are uniform.