The Dichotomy of Conjunctive Queries on Probabilistic Structures
  We show that for every conjunctive query, the complexity of evaluating it on a probabilistic database is either \PTIME or #\P-complete, and we give an algorithm for deciding whether a given conjunctive query is \PTIME or #\P-complete. The dichotomy property is a fundamental result on query evaluation on probabilistic databases and it gives a complete classification of the complexity of conjunctive queries. 
Nilesh Dalvi and Dan Suciu