An Asymptotic Probability Distribution for Almost Sure Events: A New Approximation for k n in Binomial Trials

Background: In many scientific applications, particularly in clinical trials and reliability engineering, we encounter events that occur with probability nearly one (p 1). Material and methods: The standard binomial distribution, while exact, becomes numerically challenging when the number of trials n is large and the number of successes k is close to n. The Poisson approximation, which excels for rare events (k small), is not designed for this regime. Results: This article introduces a novel probability mass function (PMF) derived from the binomial distribution under the condition k n. We provide explicit definitions of indicator random variables, Bernoulli, binomial, and Poisson distributions, including their PMFs, moments, and one-sided p-value formulas. We then derive the new PMF, prove its form, and compare its one-sided tail probabilities to the binomial and Poisson distributions using a real clinical dataset. Conclusion: The new distribution offers various advantages and new theoretical insight for nearly certain events (p 1).