A walker starts at a defined position greater than $0$, say $A$, and then makes a "decision" to walk either "$b$ steps to the right" or walk "$c$ steps to the left." He will choose the first option with probability $p$, and the second option with probability $1-p$. If the walker gets to position $0$ he stops.
I wish to calculate:
- the expectation value of the walker's position after a total of n decisions.
- what happens as n approaches infinite?
Is there an existing formula/theory that can be used to get an analytic solution to this problem?