The Mathematical Relationship Between Certainty and Probability: The Invert of The Shannon Information Entropy Curve

The relationship between certainty-level and probability-level has not been defined in mathematics literature. Some say it is pointless to define because certainty is absolute. Others imply certainty-level and confidence level are interchangeable and consequently imply certainly-level is the probability that a randomly-produced range of values called “confidence interval” contains the true value of a population parameter. Many erroneously imply certainty-level and probability-level are interchangeable. Per the following-survey question given to approximately 50 Ivy-League-graduate students, “If someone [is] 50% certain of something, what probability percentage would that mean?”—the nearly-universal answer was 50% probability. No one chose the correct answer— 89% probability. Others offer an absurd definition that would mean a ‘reasonable degree of medical/scientific certainty’ is any value greater than 0% certainty. I demonstrate the relationship is defined by the invert of the Shannon-information-entropy-curve/formula. The definitions of certainty-level (and uncertainty-level) are important to acknowledge and formalize in mathematics literature because professions that make life-and-death decisions based-on the terms have demonstrated confusion and erroneous understanding of them. If the mathematics community chooses to define certainty-level as the mathematics-community definition of confidence level, that definition would defy both the rational-language meaning of certainty-level and the meaning of the invert of the Shannon-information-entropy-formula.