Assuming a uniform random distribution:
P(A) = 0.25
P(B) = 0.25
P(C) = 0.25
P(D) = 0.25
The possible correct response sets, if multiple correct answers are possible:
P({A,D}) + P({B}) + P({C}) = 1
We do know that the answer is only correct if the selected answer is in the set of correct responses:
P(x) = P(A & {A,D}) + P(B & {B}) + P(C & {C}) + P(D & {A,D})
P(x|{A,D}) = 0.25
P(x|{B}) = 0.5
P(x|{C}) = 0.6
We can probably assume the response selection and establishing the set of correct responses are independent events, as they are being done by different people, and the person going second has no prior knowledge of the results from the first.
P(x) = (P(A)+P(D)) * P({A,D}) + P(B) * P({B}) + P(C) * P({C})
P(x) = 0.5 * P({A,D}) + 0.25 * P({B}) + 0.25 * P({C})
Unraveling:
P(x) = 0.25 * P({A,D}) + 0.5 * P({B}) + 0.6 * P({C})
P(x) = P({A,D})/4 + 1/4
P({B}) = 7/2 - 6 * P({A,D})
P({C}) = 5 * P({A,D}) - 5/2
1/2
That range doesn't overlap with the possible responses. So there's an incorrect assumption in there somewhere. Maybe the one about the question being fair. Backing up:
P({A}) + P({B}) + P({C}) + P({D}) = 1
P(x) = P(A & {A}) + P(B & {B}) + P(C & {C}) + P(D & {D})
P(x|{A}) = 0.25
P(x|{B}) = 0.5
P(x|{C}) = 0.6
P(x|{D}) = 0.25
P(x) = P(A) * P({A}) + P(B) * P({B}) + P(C) * P({C}) + P(D) * P({D})
P(x) = 0.25 * P({A}) + 0.25 * P({B}) + 0.25 * P({C}) + 0.25 * P({D})
P(x) = 0.25 * P({A}) + 0.5 * P({B}) + 0.6 * P({C}) + 0.25 * P({D})
P(x) = 0.25
0
So the correct answer is 0.25, and the correct multiple choice response is one of either A or D. One's best response is based on knowledge of the psychology of the test-writer. It is not random. The person writing the question consciously chose one or the other, intentionally violating an assumption of the test-taker--that knowing the correct answer to a question would translate to being able to unambiguously select the correct multiple-choice response.