With non-orthogonal multiple access(NOMA), we tackle the maximization of the secrecy rate for the strong user subject to a maximum allowable secrecy outage probability, while guaranteeing a constraint on the transmission rate to the weak user. For the first time, the dependence between the eavesdropper's ability to conduct successive interference cancellation and her channel quality is considered. We determine the optimal power allocation and the redundancy rate, based on which the cost of security in terms of the reduction in the strong user's secrecy rate is examined and the benefits of NOMA for secure transmissions are explicitly revealed.