Eigenvector approach for comparing the probability of an event under a pre-condition with possible incorporation of exposed population

Here, a statistical tool has been formulated to quantify the effect of a pre-condition of a probabilistic event. Main underpinning mathematical approach lies with eigenvectors of a matrix, where this matrix contains probabilities associated with an event. Perron-Frobenius theorem for positive square matrices is a key result applied in the formulation of the tool. This theorem reveals that the largest eigenvalue of a positive matrix is a positive real number and there exist a corresponding eigenvector whose components are positive real numbers. Special feature of this tool is its ability of catering possible population changes exposed under different pre-conditions of an event. Suppose a and b (> a) are the probabilities of an event subjected to a pre-condition occurs (probability w) and does not occur respectively. Here, our tool is designed to compare the probabilities a and wa + (1−w)b where more classical way of comparing them by one indicator is to consider the fraction a/wa+(1−w)b. However, new tool suggests the incorporation of exposed population fractions v 1 and v 2 to take the fraction v 1 /v 2 instead of the above classical way while satisfying the condition v 1 wa + v 2 (1−w)a/v 1 wa + v 2 (1−w)b = v 1 /v 2 . Here, left hand fraction obeys the difference in probabilities of occurring the event while right hand side imposes the dominance of exposed population. Then, one can interpret that resultant fraction is infused with three aspects: pre-condition (by w), occurrence of the event (by a and b) and population level exposure (by v 1 and v 2 ). Simplification can be approached by eigenvectors which shows a clear correspondence. If m = (wa(1−w)a/wa(1−w)b) and v = (v 1 /v 2 ), then values that associate with v 1 /v 2 must satisfy mv = λν for some λ (eigenvalue). Thereafter, Perron-Frobenius theorem and several other algebraic arguments guarantee the possible use of any positive eigenvector of the maximum eigenvalue of M to determine the wanted fraction v 1 /v 2 . Numerical values of v 1 /v 2 lie between 0 and 1 allowing good comparative ability with the classical way a/wa+(1-w)b, which also has the bounds 0 and 1. Applicability of the tool can be verified by comparing with a classical way and further extensions and modifications will arise after assimilating problem-specific situations. Sensitivity analysis can be carried out by varying one of a, b or w at a time, where differences in steepness of v 1 /v 2 allows worthwhile interpretations regarding exposed populations. This tool would be useful in recognizing the effect of pre-conditions in industrial, biological, commercial and social processes.