I'm trying to find a general method for solving problems like the following:
Flip a fair coin repeatedly, subtracting 1 if heads and multiplying by 2 if tails. If you currently have X, what is the probability that you reach, say, 1000 before reaching 0?
I've read a couple of texts on first passage problems, but haven't come across a good technique. There seem to be nice techniques for solving several similar-looking problems (for example, either divide by 2 or multiply by 3), but this version seems more difficult. Any good suggestions would be greatly appreciated!
