How are proofs verified probabilistically in interactive proof systems?
Your interpretation isn't using randomness in the right way in an MA protocol.

Every MA problem can be reduced to a polynomial-size circuit of AND-OR-NOT gates, with some input variables designated for Merlin and the rest designated for Arthur. First Merlin tells Arthur the TRUE-FALSE values of Merlin's variables. Then Arthur flips coins to choose the TRUE-FALSE values for Arthur's variables. That's all the randomness Arthur needs so he can now throw his coin away.

Now we have a logical circuit with no free variables. Arthur deterministically verifies that the circuit outputs true.