+1833 A point is on a number line, but it is limited to being at 0, 1, or 2. At each step, the point moves 1 unit left or right. If the point is at 1, it moves to either 0 or 2 with equal probability. However, if the point is at 2, it must move to 1 with its next step. For a point currently at 1, let $t_1$ be the expected number of steps the point will take before it reaches 0 for the first time. Similarly, for a point currently at 2, let $t_2$ be the expected number of steps for the point to first reach 0. Determine the ordered pair $(t_1,t_2)$.
+33615 Starting at 1 there is probability 1/2 that it moves to 0 in 1 step. There is probability 1/22 that it moves to 0 in 3 steps (1 → 2 → 1 → 0). There is probability 1/ 23 that it moves to 0 in 5 steps (1 → 2 → 1 → 2 → 1 → 0), etc.
The expected number of steps is therefore t1 = 1/2 + 3/22 + 5/23 + ... or
$$t_1=\sum_{k=1}^\infty\frac{2k-1}{2^k}=3$$
Starting at 2 it takes 1 step to 1 from where the expected number of steps is 3 as obtained above, so t2 = 1+3 or t2 = 4
So the ordered pair is (t1, t2) = (3, 4)
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+33615 Starting at 1 there is probability 1/2 that it moves to 0 in 1 step. There is probability 1/22 that it moves to 0 in 3 steps (1 → 2 → 1 → 0). There is probability 1/ 23 that it moves to 0 in 5 steps (1 → 2 → 1 → 2 → 1 → 0), etc.
The expected number of steps is therefore t1 = 1/2 + 3/22 + 5/23 + ... or
$$t_1=\sum_{k=1}^\infty\frac{2k-1}{2^k}=3$$
Starting at 2 it takes 1 step to 1 from where the expected number of steps is 3 as obtained above, so t2 = 1+3 or t2 = 4
So the ordered pair is (t1, t2) = (3, 4)
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