Question

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

Answer 1

Complete question is;

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)

Answer:

(e_1,e_2) = (2,3)

Step-by-step explanation:

We can track the probabilities using the scenario that Ant gets to 0 after 1 steps, 2 steps, 3 steps, 4 steps, 5 steps and so on.

This gives us e_1 = 1(1/2^(1)) + 2(1/2²) + 3(1/2³) + 4(1/2⁴) + ...

By arithmetico-geometric series, which is given by;

S_(∞) = [a/(1 - r)] + dr/(1 - r)²

From the online calculator, i got;

e_1 = 2

Similarly, e_2 = 2(1/2^(1)) + 3(1/2²) + 4(1/2³) + 5(1/2⁴) + ...

By arithmetico-geometric series, e_2 = 3,

Thus, (e_1,e_2) = (2,3).