I was just exposed to this puzzle:

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%

My answer is: 25%. But here’s just a stab at an explanation (which makes this puzzle seem similar to Newcomb’s paradox).

Either A or D is the correct answer, and your choice determines which is correct (and, consequently, which is incorrect). My initial thought is that counterfactual semantics may be of some help.

Assume there can only be one correct answer in any possible world in which an answer is chosen. So for any world in which A is chosen, A is the correct answer in that world; likewise, any world in which D is chosen, D is the correct answer in that world.

Suppose you choose A in this world (the actual world, call it *W*). A is thereby the correct answer in *W*, and D is the incorrect answer. But D is correct in another possible world (say, *W**) and A, incorrect. I.e., the following counterfactual is true: Had *W** been actual, you would have chosen D, and D would have been the correct answer (and A would have been the incorrect answer). But it does *not* follow that D could have been the correct answer in *W*, because there can only be one correct answer per world, and you chose A in *W*. In other words, which answer is correct is logically posterior your choice, and that there is no correct answer prior to your choosing. That last bit may sound odd, but isn’t this exactly what we should expect if odds can be determined by random choice (i.e., there is no arbitrary choosing, there are no odds of choosing correctly)?

But I must confess that, probably, the chances of me being correct in my explanation are probably less than 25%. Enter the Paradox of the preface.

There seem to be two ways of looking at this problem.

Either we can become bogged down in the language of the question and conclude that the problem is poorly posed and ambiguous and therefore cannot be ansered. This seems like the easy way out to me and I think we can do better. If we actually attempt to answer the question we quickly see that it is ‘self-referential’ and for any answer to be correct it would need to be ‘self-consistent’. Let’s first consider a similar (simpler) problem with slightly different answer options to establish what I mean by ‘self-consistent’ and settle on a criteria for determining a ‘correct’ answer. If the available answer options were …

A) 20%

B) 10%

C) 40%

D) 40%

E) 50%

… we can examime the four available answer values to see whether any of them are, in fact, ‘self-consistent’.

There is a 20% chance you will randomly select an answer of 10%

There is a 20% chance you will randomly select an answer of 20% (and 20% = 20% so this is ‘self-consistent’)

There is a 40% chance you will randomly select an answer of 40% (and 40% = 40% so this is ‘self-consistent’)

There is a 20% chance you will randomly select an answer of 50%

So, in the simplified question above, I would consider the answers A, C, and D to all be ‘correct’. Let’s return now to the orginal question and apply the same logic. Remember, our options were …

A) 25%

B) 50%

C) 0%

D) 25%

… and again we can examine the three availabe answer values to see whether any of the are ‘self consistent’.

There is a 25% chance you will randomly select an answer of 0%.

There is a 50% chance you will randomly select an answer of 25%.

There is a 25% chance you will randomly select an answer of 50%.

It seems that none of these answer values are ‘self-consistent’. In no case is the value of the answer equal to the likelihood of you randomly selecting that answer. The problem does not have a valid solution. In my opinion this is a truly interesting paradox – but not a problem to be solved.

Comment by oddsandprobs — October 23, 2012 @ 1:19 am |

Sure, feel free to pass on my comments to whoever you like.

But don’t read too deeply into what I have written because I don’t know very much about logic or philosophy. The terms ‘self-referential’ and ‘self-consistent’ are phrases up because I was having trouble expressing what I meant.

I guess that what I meant by ‘self-consisten’ is that if we assume that a particular answer is correct then this must not lead to any conclusions that contradict our original assumption. This would not be ‘self-consistent’. In the case of this particular problem, if we assume a particular answer is correct – say 25% – then there must actually be a 25% chance that you will select this answer. Otherwise our assumption has lead to a conclusion that contradicts our original assumption and that original assumption must, therefore, be ‘incorrect’.

Feel free to pass this however you see fit. Credit or a link to my blog would be nice though …

Comment by oddsandprobs — October 23, 2012 @ 2:51 am |