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 \( k = n - Xobs \) 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 \( H_0: p ≈ 1 \). Let \( X \) denote the number of successes, with observed value \( Xobs = n - k \), where \( k \) is small. The exact right-tailed \( p \)-value is given by\(p-value = ∑ⱼ₌₀ᵏ {n}{j} p\,n-j (1-p)ʲ.\)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 \( p̂ = k/n ≈ 1 \), we obtain the approximation\(p-value ≈ p̂\,k.\)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 \( p̂\,k \). 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.