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!

JustAPotato Mar 6, 2020

#1**0 **

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).

Guest Mar 6, 2020