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We introduce ARB-Dropout, an efficient single-pass alternative to Monte Carlo (MC) Dropout for uncertainty estimation in deep neural networks. ARB-Dropout adaptively determines per-input dropout rates from gradient variance and analytically propagates the resulting epistemic variance through the network, eliminating the need for multiple stochastic forward passes. By integrating this analytic epistemic estimate with heteroscedastic aleatoric noise from a dedicated output head, ARB-Dropout produces well-calibrated predictive distributions with lower Expected Calibration Error (ECE) and Negative Log-Likelihood (NLL) than MC Dropout, while maintaining comparable accuracy. Experiments on CIFAR-10, CIFAR-100, SVHN, and STL-10 demonstrate that ARBDropout offers substantial inference-time speedups and robust uncertainty estimates, making it well-suited for real-time, uncertainty-aware applications.