Is there any notion of sensitivity for probabilistic Boolean functions?
This should work fine inasmuch as if $f$ is now a random function, then you just get the expected sensitivity:
$$\mathbb E s(f, x) = \sum_{y \in N(x)} \mathbb E I(f(x) \neq f(y)) = \sum_{y \in N(x)} \Pr(f(x) \neq f(y)).$$