  We study formal languages which are capable of fully expressing quantitative
probabilistic reasoning and do-calculus reasoning for causal effects, from a
computational complexity perspective. We focus on satisfiability problems whose
instance formulas allow expressing many tasks in probabilistic and causal
inference. The main contribution of this work is establishing the exact
computational complexity of these satisfiability problems. We introduce a new
natural complexity class, named succ$\exists$R, which can be viewed as a
succinct variant of the well-studied class $\exists$R, and show that the
problems we consider are complete for succ$\exists$R. Our results imply even
stronger algorithmic limitations than were proven by Fagin, Halpern, and
Megiddo (1990) and Moss\'{e}, Ibeling, and Icard (2022) for some variants of
the standard languages used commonly in probabilistic and causal inference.
