The paper considers the problem of analyzing the randomized error in the Lp sense for eigenvalue and eigenvector estimate by methods based on Krylov information. We report bounds for the approximation to the dominant eigenvalue of a real symmetric matrix by the power method, we then generalize those bounds to normal matrices. We show that we can apply upper bounds on the randomized error for estimating the dominant eigenpair by the Lanczos method in order to compute an approximation to the smallest eigenpair. We then give a randomized algorithm for computing the condition number and we study its randomized error.