In this work we analyze the sensitivity of indicator random variables associated with a failure event that occurs with a small probability. This is a classical problem in engineering risk and reliability analysis. For the sensitivity analysis we use the well known Sobol' indices defined in terms of the variance decomposition of the failure indicator random variable. We derive tight two-sided bounds for the closed and total Sobol' indices of affine linear limit-state functions and standard Gaussian inputs. In addition we show that the closed and total Sobol' indices approach either zero or one in the limit of the failure probability going to zero. This proves rigorously an empirical observation in previous studies. Moreover, we study second-order approximations of reliability sensitivity indices and show their asymptotic behavior under certain conditions. We illustrate the theoretical results by a simple test problem with an affine linear limit-state function. Moreover, we consider a practical engineering test problem with a nonlinear limit-state function.