Probability and variance

Abstract

Background: The highly conservative nature of Chebyshev's inequality, with its advantages and disadvantages, calls for an additional approach to the event probability of an individual event across various distributions. Materials and Methods: This investigation commences with the theoretical foundations of event probability, presenting a mathematical framework for the precise quantification of event probability independent of the distribution of the random variable. Simultaneously, we explore the relationship between the event probability of an individual event and the variance of that event. Results: Chebyshev's inequality can function more or less as a rough upper limit for event probability, depending on the number of standard deviations from the mean. The results of this study suggest that the variance and the expected value of an event allow for a very precise determination of the event probability of an individual event, eliminating the need for estimation. Conclusion: In summary, this study sheds light on the dynamic relationship between variance and event probability, emphasizing the limitations of Chebyshev's inequality. This insight can prove particularly valuable in scenarios with non-normally distributed datasets.