Fri 21 May 2010

## A self-referential multiple choice puzzle

Posted by Happy under puzzle

Comments Off on A self-referential multiple choice puzzle

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.

Trial 4. Answer D: same logic as answer A. Fail.

Result: 4 fails, there is no chance of picking the correct result by chance.

You might think by inspection that this outcome would make Answer C correct, but it does not. When you try to choose it in a trial by this careful procedure you find out that Answer C is actually incorrect, like all the others. Appearances can be deceptive. The paradox is that there appears to be a correct answer, and yet you can never choose it.

The point here is that this is a failed paradox. There is no “correct” answer to this question ante facto, because marking takes place after an answer has been selected. The reason why that is so is because we don’t know the answer until after the selection is made. The consequence is that the chance of making a correct selection is nil, even if you make selection C (0%).

I did a bit of Googling, but couldn’t find any authoritative analysis. Just lots of chatter.