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 \(\sigma\)-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 $(p_L)$ and upper $(p_U)$ confidence bounds\(p_L \le p_U,\)with\(p_U = p_L + \Delta,\)where\(\Delta\)denotes the confidence width.Results:For events considered certain,\(p = p_U = 1.\)By identifying\(\Delta = \alpha,\)where\(\alpha \in (0,1)\)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\(p_L = 1-\alpha.\)The right-tailed p-value for observing $(p_L)$ in\(n\)independent trials becomes\(p\text{-value}_{\mathrm{right}}:=(1-\alpha)^n.\)Conclusion:We show that under conditions, where the empirical lower confidence bound is equal to\(p_L = 1-\alpha,\)then it is reasonable to accept that the underlying $p\text{-value}$ satisfies\(p\text{-value}_{\mathrm{right}}:=(1-\alpha)^n.\)