The probability that an employee selected at random from the company is a junior employee who favors the policy is 1/8.…

Let `F` be the event that a randomly selected employee favors the policy, `J` be the event that the employee is junior, and `S` be the event that the employee is senior. The total number of employees is `j + s`.

The number of junior employees who favor the policy is `j · p`. The probability of selecting a junior employee who favors the policy is the number of junior employees who favor the policy divided by the total number of employees. This is given as 1/8.

`P(J and F) = (j · p) / (j + s) = 1/8`

This is the first key step. We must solve this equation for `p` to use later. Multiplying both sides by `(j + s)` gives `j · p = (j + s) / 8`. Dividing by `j` gives `p = (j + s) / (8j)`.

The overall probability that a randomly selected employee favors the policy, `P(F)`, is the sum of the probabilities of selecting a junior who favors and selecting a senior who favors.

`P(F) = P(J and F) + P(S and F)`

The number of senior employees who favor the policy is `s · (1 - p)`.

So, `P(S and F) = (s · (1 - p)) / (j + s)`.

Therefore, `P(F) = 1/8 + (s · (1 - p)) / (j + s)`.

Now, substitute the expression for `p` into this equation:

`P(F) = 1/8 + (s / (j + s)) · (1 - (j + s) / (8j))`

Simplify the term in the parentheses:

`1 - (j + s) / (8j) = (8j / 8j) - (j + s) / (8j) = (8j - j - s) / (8j) = (7j - s) / (8j)`.

Substitute this back into the expression for `P(F)`:

`P(F) = 1/8 + (s / (j + s)) · ((7j - s) / (8j))`

`P(F) = 1/8 + (s(7j - s)) / (8j(j + s))`

To add these fractions, find a common denominator, which is `8j(j + s)`. Rewrite 1/8 with this denominator:

`1/8 = (j(j + s)) / (8j(j + s))`

Now add the two parts:

`P(F) = (j(j + s)) / (8j(j + s)) + (s(7j - s)) / (8j(j + s))`

`P(F) = (j² + js + 7js - s²) / (8j(j + s))`

`P(F) = (j² + 8js - s²) / (8j(j + s))`

Therefore, the correct choice is A.

B is incorrect because it results from a sign error when simplifying `1 - (j + s) / (8j)`, calculating it as `(8j - j + s) / 8j`. This leads to `+ s²` in the numerator instead of `- s²`.

C is incorrect because it results from the misconception that `p = 1/8`, confusing the joint probability `P(J and F)` with the conditional probability `P(F|J)`. Substituting `p=1/8` into the general formula `(jp + s(1-p)) / (j+s)` yields this distractor.

D is incorrect because it results from incorrectly combining probabilities, adding the probability of a junior favoring the policy (`1/8`) to the probability that a senior favors the policy (`1-p`), without weighting the second probability by the proportion of senior employees. This calculates `1/8 + (1-p)`.