Problem:
A moving particle starts at the point and moves until it hits one of the coordinate axes for the first time. When the particle is at the point , it moves at random to one of the points , or , each with probability , independently of its previous moves. The probability that it will hit the coordinate axes at is , where and are positive integers, and is not divisible by 3 . Find .
Solution:
All paths that first hit the axes at the origin must pass through the point . There are paths from the point to the point that take steps left and steps down, that take steps left, steps down, and diagonal step, that take step left, step down, and diagonal steps, and that takes diagonal steps. The total probability of moving from to is therefore
Multiplying by gives , the probability that the path first reaches the axes at the origin. The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions