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 e_1 be expected number of steps Fred takes to get to 0, given that Fred starts at the point 1. Similarly, let e_2 be expected number of steps Fred takes to get to 0, given that Fred starts at the point 2. Determine the ordered pair (e_1,e_2). Answer is NOT (2, 3)

The ordered pair (e₁, e₂) is (6, 8).

Step-by-step explanation:

Consider the pathway attached below.

It is provided that Fred moves to either 0 or 2 with equal probability, i.e. 0.50.

    e₁ :    1              3             5              7        ...

P (e₁) : 0.50       0.50²      0.50³       0.50⁴   ...

The expected number of steps Fred takes to get to 0 if he is at 1 is:

The sum series e₁ is an AGP.

The sum of infinite AGP is:

Then the value of e₁ is:

It is provided that Fred always moves to 1 if he at step 2.

    e₂ :    2             4             6              8        ...

P (e₂) : 0.50       0.50²      0.50³       0.50⁴   ...

The expected number of steps Fred takes to get to 0 if he is at 2 is:

The sum series e₂ is an AGP.

The sum of infinite AGP is:

Then the value of e₂ is:

Thus, the ordered pair (e₁, e₂) is (6, 8).

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.

7,24

step-by-step explanation: