From One to All and Back: Theorems on Probability

Background: This paper establishes two fundamental theorems relating the per-trial probability p of a binary event to the observed probability \ (p₂₀₋₂\) of observing that event in all n independent trials. Material and methods: The theoretical framework is validated through both population-level and sample-level derivations. Results: The first theorem states that the probability of observing the event in every one of n independent Bernoulli trials is \ (p^n\). The second theorem provides the inverse relationship, showing that the per-trial probability is the n-th root of the observed joint probability. The theorems are then applied to the evaluation of necessary condition relationships in empirical data, demonstrating that a perfect empirical fit requires equality between the observed proportion and the theoretical population proportion. Conclusion: These results are foundational for any statistical analysis involving repeated independent binary outcomes and are presented with formal proofs.