Here's a good brain puzzler:

[Nathan G]

If I choose an answer at random, what's the probability of getting it correct?

A) 25%

B) 0%

C) 50%

D) 25%

 

[Englund]

Well, I am not sure what the percents means. But if there are four answers where one of them is right, the probability of getting the correct answer when choosing at random is 1/4.

 

[TheEcologist]

Yeah, but there are two 1/4 in the answer hence one could argue 1/2, but since the answer 1/2 itself is only one of 4, choosing purely randomly would lead you back to a prob of 1/4.

It's a troll question

 

[Nathan G]

It's a troll question

I guess I owe you guys an apology. I'm sorry. I am actually interested if there is any way around the paradoxical nature of the question. Something so seemingly simple can have devastating implications. Does making a choice actually change the answer? If so, is there a convergence somewhere in between as the number of choices go to infinity? Please, don't pop any brain cells over it, I was just wondering if anyone knew off hand. To the professors out there, this would be truly diabolical to put on a quiz.

 

[Dason]

I guess I owe you guys an apology. I'm sorry. I am actually interested if there is any way around the paradoxical nature of the question. Something so seemingly simple can have devastating implications. Does making a choice actually change the answer? If so, is there a convergence somewhere in between as the number of choices go to infinity? Please, don't pop any brain cells over it, I was just wondering if anyone knew off hand. To the professors out there, this would be truly diabolical to put on a quiz.

If we assume that choosing an answer at random means randomly choosing one of the 4 answers according to a uniform distribution then there just is no consistent solution. That's about all there is to say.

However... this problem does become a little more (to me) when you ask what probability distribution over the four answers leads to a situation where there is a correct answer. For that it makes a difference if you want to say there is only one correct letter answer (only a, b, c, or d will be marked as correct) or if you want to say the user just needs to choose a correct answer (so as long as they choose an answer that has a correct response it gets marked as correct).