If you are at 0 or 1.

Say your initial position is at $0.7$... You are closest to 1. So $d$ is $0.3$. You have a 50% chance of ending at $1$, 50% chance of ending at $0.4$. If the latter, repeat. If the former, the game ends. Repeat means here $d$ is 0.4 because closest to zero. 50% chance of ending at 0 (ending the game, but unsuccessfully), 50% chance of moving to 0.8. Repeat until you end at 0 or 1. The question is, what is the probability you end at 1.