How are proofs verified probabilistically in interactive proof systems? I'm having a hard time understanding the way Arthur verifies proofs probabilistically with coin tosses in an intuitive manner.

Suppose Arthur is a logician equipped with paper, a pencil and an unbiased coin. Someone hands Arthur a proof that some string s is in some language L. How exactly does Arthur probabilistically check the proof? My interpretation is that he does the following: 

Arthur tosses the coin. If it lands showing heads, he accepts the first statement without checking it. If it lands tails, he checks it. If, upon checking, he determines that the first statement is false, he rejects the proof. Otherwise, if he determines that the first statement is true, he continues this process with the next statement of the proof. If Arthur never rejects the proof during this process, then he accepts the proof if one of the statements is a concluding statement of the form "..., hence s is in L" and he rejects the proof otherwise.

Is this a correct interpretation?

Edit: I am not talking about probabilistically checkable proofs. The question above came about from thinking about the class MA.
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.