Multiple Choice Question:
If you choose an answer to this question at random, what is the chance you will be correct?
A: 25%
B: 50%
C: 0%
D: 25%

The answer is none. You will not get a correct answer by choosing at random.

The problem is set up to look like a typical liar’s paradox and it gets close, but not this time. The explanation goes something like this.

There are only two ways I know to get accurate answers for questions of this kind that depend on probabilities.
1. Enumerate the cases and count the results.
2. Turn it into maths, and apply standard theorems and transformations.

Assuming we’re not mathematicians, enumerating the cases is the way to go.

There are 4 equally probably choices that could be made: A,B,C,D. We treat each choice as a separate trial which either succeeds (the answer is correct) or fails (the answer is incorrect); there are no fractional results for a trial. The required overall probability of a correct answer is the number of successful trials divided by 4.

Trial 1: Answer A is correct if and only if it is correct and all other answers are incorrect. However, answers A and D are identical so if answer A is correct then answer D must also be correct. Therefore answer A is wrong. Fail.

Trial 2. Answer B is correct if it is correct and exactly one other answer is correct. However, if answer B is correct then answers A, C and D must be incorrect. Therefore answer B is incorrect. Fail.

Trial 3. Answer C is correct if no answers are correct. Answer C cannot be both correct and incorrect, therefore answer C must be incorrect (in which case there is no contradiction). Fail.