  Shapley values, originating in game theory and increasingly prominent in
explainable AI, have been proposed to assess the contribution of facts in query
answering over databases, along with other similar power indices such as
Banzhaf values. In this work we adapt these Shapley-like scores to
probabilistic settings, the objective being to compute their expected value. We
show that the computations of expected Shapley values and of the expected
values of Boolean functions are interreducible in polynomial time, thus
obtaining the same tractability landscape. We investigate the specific
tractable case where Boolean functions are represented as deterministic
decomposable circuits, designing a polynomial-time algorithm for this setting.
We present applications to probabilistic databases through database provenance,
and an effective implementation of this algorithm within the ProvSQL system,
which experimentally validates its feasibility over a standard benchmark.
