Truncated Gaussian Approximations for Near-Certain Binomial Events: Bridging the Gap to Poisson Limits.

Background:In experimental settings with extremely high success probabilities, classical probabilistic models encounter limitations in accurately describing tail behavior. In particular, when the number of observed successes approaches the theoretical maximum, standard Gaussian approximations to the binomial distribution become unreliable.Material and methods:This work introduces a truncated Gaussian framework that explicitly accounts for the natural upper bound of binomial outcomes. Starting from the de Moivre--Laplace approximation, we derive a properly normalized truncated density and establish its analytical properties. The approach is compared to both the classical Gaussian approximation and the Poisson limit for rare events.Results:We demonstrate that while the Gaussian approximation performs well near the mean, it systematically underestimates extreme tail probabilities. The truncated Gaussian improves structural consistency but does not fully resolve this limitation. In contrast, the Poisson approximation provides accurate estimates in the rare-event regime.Conclusion:These findings highlight the importance of selecting appropriate asymptotic models for inference in high-success settings and provide a unified framework linking binomial, Gaussian, and Poisson behavior.