Anty the ant is on the real number line, and Anty's goal is to get to $0.$ If Anty is at $1,$ then on the next step, Anty moves to either $0$ or $2$ with equal probability. If Anty is at $2,$ then on the next step, Anty always moves to $1.$ Let $e_1$ be expected number of steps Anty takes to get to $0,$ given that Anty starts at the point $1.$ Similarly, let $e_2$ be expected number of steps Anty takes to get to $0,$ given that Anty starts at the point $2.$ Determine the ordered pair $(e_1,e_2)$.
You can track the probabilities using different case, i.e. Anty gets to 0 after 1 steps, 2 steps, 3 steps, and so on. This gives us e_1 = 1/2*1 + 1/4*2 + 1/8*3 + 1/16*4 + ... By arithmetico-geometric series, e_1 = 2. Similarly, e_2 = 1/2*2 + 1/4*3 + 1/8*4 + 1/16*5 + ... = 3, so (e_1,e_2) = (2,3).
