A particle moves at each step two units to the right or one unit to the left, with corresponding probabilities p and q (p + q = 1). If the starting position is 0, find the probability

a, that the particle will ever reach the origin. (This is the ruin problem against an infinitely rich adversary.) Hint: The equation corresponding to (2.1) has the particular solution 92 = 1 and two particular solutions of the form , where a satisfies a quadratic equation.