<p>Multi-criteria decision-making problems in critical decision environments often suffer from two persistent challenges: experts often provide imprecise judgments, and classical AHP cannot adequately propagate this uncertainty. Existing stochastic or fuzzy variants help but still rely on pointwise simplification, heavy simulation, or ignoring compositional constraints. This paper introduces the Stochastic Analytic Hierarchy Process (SAHP), a closed-form Bayesian framework that integrates Beta-PERT triplet elicitation with a log-ratio Gaussian model to produce uncertainty-aware priority weights and ranking probabilities. SAHP directly transforms min-mode-max expert judgments into moment-based likelihoods, performs inference on a reduced-dimension log-weight space, and yields analytically tractable posterior distributions. This leads to a fast, transparent, and operationally feasible pipeline for uncertainty-propagating AHP. The framework is applied to a real-world ICU prioritization problem using clinical data from three COVID-19 waves. Results demonstrate that SAHP captures temporal shifts in clinical importance, produces stable and interpretable priority rankings, and offers clear uncertainty diagnostics through posterior intervals and rank probabilities. Comparative analyses show full agreement with classical AHP rankings under consistent inputs, while sensitivity tests confirm exceptional robustness to perturbations in elicited weights. The proposed SAHP advances the state of probabilistic MCDM by unifying practitioner-friendly elicitation, statistical rigor, and computational efficiency. The method is broadly applicable to healthcare, disaster response, infrastructure allocation, and any decision environment requiring rapid, explainable, and uncertainty-aware prioritization.</p>