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Fred the ant is on the real number line, and Fred is trying to get to the point 0

If Fred is at  1 then on the next step, Fred moves to either  0 or   2 with equal probability. If Fred is at 2  then on the next step, Fred always moves to 1.

Let e1 be expected number of steps Fred takes to get to 0 given that Fred starts at the point 1. Similarly, let  e2 be expected number of steps Fred takes to get to 0  given that Fred starts at the point 2.

Determine the ordered pair (e1,e2).

 Jun 6, 2020
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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).

 Jun 6, 2020

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