Binomial Probabilities Under the Null Hypothesis

Background. In hypothesis testing involving binomial proportions, the conventional \ (p \) -value calculation often requires summation over many terms. However, when the null probability \ (p \) is extremely close to \ (1 \) and the observed number of failures \ (k = n - X₎₁ₒ \) is very small, the right-tailed \ (p \) -value simplifies considerably. Material and methods. We consider a binomial setting with \ (n \) independent trials under the null hypothesis \ (H₀: p 1 \). Let \ (X \) denote the number of successes, with observed value \ (X₎₁ₒ = n - k \), where \ (k \) is small. The exact right-tailed \ (p \) -value is given by\ (p-value = ₉=₀^k nj p^\, n-j (1-p) ^j. \) For \ (p \) near \ (1 \) and small \ (k \), we analyze the dominant term in this sum. Results. The dominant contribution arises from the term with \ (j = 0 \) (no failures), yielding\ (p-value p^\, n. \) Expressing this in terms of the observed failures \ (k \) and the observed success rate \ (p = k/n 1 \), we obtain the approximation\ (p-value p^\, k. \) This approximation becomes increasingly accurate as \ (p \) approaches \ (1 \) and \ (k \) remains small. Conclusions. When testing a null hypothesis with \ (p \) very close to \ (1 \) and observing only a few failures (\ (k \) small), the right-tailed \ (p \) -value can be reliably approximated by \ (p^\, k \). This simple formula provides a practical alternative to full binomial summation, facilitating rapid interpretation in high-reliability contexts such as pharmacovigilance, quality control, and rare event analysis.