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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 \((e_1,e_2)\).

 Dec 14, 2019
 #1
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(e_1, e_2) = (4,6).

 Dec 15, 2019
 #2
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Do you have proof for this?

Guest Dec 15, 2019
 #3
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I'm getting (e_1, e_2) = (1,2).

 Dec 17, 2019

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