You can't trick a theorem by adding randomness. It's not like a person who can be surprised by the direction you went and admit, "oh maybe I hadn't thought of that".

No, but sometimes you can use randomness as a loophole if a theorem doesn't apply to the randomized situations. And when that does and does not help is often not clear. That's why for example it is a deep open problem whether in computational complexity randomness gives you anything new. There are two generalizations of polynomial time using randomness. One is BPP and is comparatively small, and is believed to probably be equal to P. But with only a tiny tweaking of the definition, one gets the class PP which is known to be large enough to at least contain all of NP. So if P != NP, then this is a context where randomness really does help. (Although note that some people see this as an argument for believing that P=NP since their intuition is strongly that randomness shouldn't help.)