Following a review of statistical models of the failure of single fibers and bundles of these fibers, algebraic recurrence formulas are derived that generate expressions for the failure probabilities of bundles of classical fibers. One of these recurrence relations is suitable for the accurate numerical calculation of failure probabilities of bundles consisting of up to 500 single fibers. It is shown how account can be taken of the effect of defect-free fibers having finite strength. Numerical results are compared to three asymptotic analytic approximations, two of which have been proposed in the statistical literature and are now applied to fiber problems for the first time.