They don’t say that the random answer is chosen uniformly (though that is the norm in the field). If we relax that, then we’re putting a distribution on these where we want:

P(correct with distribution (a,b,c,d)) = some value shown on A,B,C,D

I don’t see any assumption that we will pick using that distribution, so I think this avoids the recursion.

Unfortunately this has too many solutions. If you put a total of 0.25 weight on A and D, then the rest does not matter. If you put 0.5 weight on C, again the rest is irrelevant.

Wow that would kill a lot of people where I grew up. Very efficient.