In this article, we derive a closed-form expression for the probability of successful packet transmission (PSPT) over Rayleigh fading channels in cognitive radio networks (CRNs). The derived expression is novel, as it does not require prior channel state information. This is achieved by employing a probabilistic model for the packet transmission time, avoiding the less realistic deterministic model that is widely used in the literature. Furthermore, the derived expression allows trading accuracy with computational complexity. Our expression is computationally efficient with linear computational complexity. Simulation results demonstrate high accuracy in evaluating the PSPT, with controllable computational complexity.