  A bistochastic matrix B of size N is called unistochastic if there exists a
unitary U such that B_ij=|U_{ij}|^{2} for i,j=1,...,N. The set U_3 of all
unistochastic matrices of order N=3 forms a proper subset of the Birkhoff
polytope, which contains all bistochastic (doubly stochastic) matrices. We
compute the volume of the set U_3 with respect to the flat (Lebesgue) measure
and analytically evaluate the mean entropy of an unistochastic matrix of this
order. We also analyze the Jarlskog invariant J, defined for any unitary matrix
of order three, and derive its probability distribution for the ensemble of
matrices distributed with respect to the Haar measure on U(3) and for the
ensemble which generates the flat measure on the set of unistochastic matrices.
For both measures the probability of finding |J| smaller than the value
observed for the CKM matrix, which describes the violation of the CP parity, is
shown to be small. Similar statistical reasoning may also be applied to the MNS
matrix, which plays role in describing the neutrino oscillations. Some
conjectures are made concerning analogous probability measures in the space of
unitary matrices in higher dimensions.
