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 x be expected number of steps Anty takes to get to 0 given that Anty starts at the point 1. Similarly, let y be expected number of steps Anty takes to get to 0 given that Anty starts at the point 2.
Determine the ordered pair (x,y).
So so far I've figured that if he gets to 0 on the first try, the probability is 1/2. 2nd try = 1/8. 3rd try = 1/32 etc etc. I have no idea how to figure expected value with an infinite number, can someone help me figure this cool problem out please? Thank you!
That's right, you can compute the probabilities, and add them up! 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).