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 be expected number of steps Anty takes to get to 0, given that Anty starts at the point 1. Similarly, let be expected number of steps Anty takes to get to 0, given that Anty starts at the point 2. Determine the ordered pair .
There qwas the same question, but the answers were wrong.
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).
