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This only produces a paradox if you fall for the usual fallacy that "at random" necessarily means "with uniform probability".
For example, I would pick an answer at random by rolling a fair cubic die and picking a) if it rolls a 1, b) on a 2, d) on a 3 or c) otherwise so for me the answer is c) 50%.
However, as it specifies that you are to pick at random the existence, uniqueness and value of the correct answer depends on the specific distribution you choose.
33% innit
It is 33% if the answer itself is randomly chosen from 25%, 50%, and 60%. Then you have:
If the answer is 25%: A 1/2 chance of guessing right
If the answer is 50%: A 1/4 chance of guessing right
If the answer is 60%: A 1/4 chance of guessing right
And 1/3*1/2 + 1/3*1/4 + 1/3*1/4 = 1/3, or 33.333...% chance
If the answer is randomly chosen from A, B, C, and D (With A or D being picked meaning D or A are also good, so 25% has a 50% chance of being the answer) then your probability of being right changes to 37.5%.
This would hold up if the question were less purposely obtuse, like asking "What would be the probability of answering the following question correctly if guessing from A, B, C and D randomly, if its answer were also chosen from A, B, C and D at random?", with the choices being something like "A: A or D, B: B, C: C, D: A or D"
iis
100 **** percent, i'm all in!
Since two of them are the same, you have a 50% chance of picking something that is 33% of the possible answers. The other two, you have 25% chance of picking something that us 33% of the possible answers.
So 50%33% + 2 (33%*25%)= 33%
So your chances of being right is 33% cause there is effectively 3 choices.
I argue it's still 25%, because the answer is either a,b,c, or d, you can only choose 1, regardless of the possible answer having two slots.
Yup. And it says pick at random. Not apply a bunch of bullshit self mastubatory lines of thinking. Ultimately, 1 of those answers are keyed as correct, 3 are not. It's 25% if you pick at random. If you're applying a bunch of logic into it you're no longer following the parameters anyway.
You can just say "I don't understand probability (or the word 'if')" next time and save a whole bunch of effort.
Ahhhhhhhhhhhhhhhhh
There's a reason I dropped probability at school.
Any answer is correct as long as you don't pick it at random. I'd choose (a) because I'm too lazy to read the other options