Beyond the Rule of Three: A Simple Lower Confidence Bound for Certain Events

Background:Classical confidence intervals are traditionally formulated within the framework of measure theory, probability spaces, and \(σ\)-algebras.In the present work, we introduce a simplified approach to confidence bounds based on lower and upper confidence levels.Material and Methods:The proposed approach avoids explicit dependence on measure-theoretic foundations and instead presents confidence bounds as a relation between repeated observations and admissible parameter regions. In general, we define lower (pL) and upper (pU) confidence bounds\(p_L ≤ p_U,\)with\(p_U = p_L + Δ,\)where\(Δ\)denotes the confidence width.Results:For events considered certain,\(p = p_U = 1.\)By identifying\(Δ = α,\)where\(α ∈ (0,1)\)denotes the significance level of the underlying statistical test, representing the maximal tolerated probability of falsely rejecting the null hypothesis, the lower confidence bound becomes\(p_L = 1-α.\)The right-tailed p-value for observing (pL) in\(n\)independent trials becomes\(p-valueright:=(1-α)^n.\)Conclusion:We show that under conditions, where the empirical lower confidence bound is equal to\(p_L = 1-α,\)then it is reasonable to accept that the underlying p-value satisfies\(p-valueright:=(1-α)^n.\)