  For a large class of C*-algebras $A$, we calculate the $K$-theory of reduced
crossed products $A^{\otimes G}\rtimes_rG$ of Bernoulli shifts by groups
satisfying the Baum--Connes conjecture. In particular, we give explicit
formulas for finite-dimensional C*-algebras, UHF-algebras, rotation algebras,
and several other examples. As an application, we obtain a formula for the
$K$-theory of reduced C*-algebras of wreath products $H\wr G$ for large classes
of groups $H$ and $G$. Our methods use a generalization of techniques developed
by the second named author together with Joachim Cuntz and Xin Li, and a
trivialization theorem for finite group actions on UHF algebras developed in a
companion paper by the third and fourth named authors.
