  We apply the operation of random independent thinning on the eigenvalues of
$n\times n$ Haar distributed unitary random matrices. We study gap
probabilities for the thinned eigenvalues, and we study the statistics of the
eigenvalues of random unitary matrices which are conditioned such that there
are no thinned eigenvalues on a given arc of the unit circle. Various
probabilistic quantities can be expressed in terms of Toeplitz determinants and
orthogonal polynomials on the unit circle, and we use these expressions to
obtain asymptotics as $n\to\infty$.
