Expert opinions, fuzzy probabilities, and warnings: toward a mathematical method for evaluating warning “read and heed” effectiveness

In many situations, and particularly in legal matters, decisions may be based on whether some possibility is judged to be “more probable than not”, as for example, whether a warning, had it been present, would “more probably than not” have prevented an injury. The probability of a warning accomplishing its purpose of preventing injury can, in principle, be computed based on probability calculus. However, this paper proposes that: (1) such computations must use “fuzzy” probability statements which as a result produce a range of weighted possibilities for the final outcome and, moreover; (2) the critical statistic for determining whether the outcome is “more probable than not” is the median rather than the mean of the possible outcomes. This paper briefly outlines the mathematics for such computations and provides several examples. The paper basically treats fuzzy probabilities as standard “random variables” restricted to the range of 0,1].For brevity, this paper only treats the multiplication of two or more fuzzy probabilities. However, this is adequate for evaluation of a two stage “read and heed”, or of longer multiple step linear sequences and is indicative of the general approach. Treatment of the additive and general case will be found in later publications.