The problem may have a simple answer, but 0.5 is not it, because the answer will be a function of n, the starting position. If n is near 0, the chance you hit

The problem may have a simple answer, but 0.5 is not it, because the answer will be a function of n, the starting position. If n is near 0, the chance you hit zero before 1000 steps is larger. The keyword is "random walk hitting time".