Minimizing the Average Number of Inspections for Detecting Rare Items in Finite Populations

Frequently one has to search within a finite population for a single particular individual or item with a rare characteristic. Whether an item possesses the characteristic can only be determined by inspection. The availability of additional information about the items in the population opens the way to more effective inspection than just random or complete inspection of the population. We will assume that the available information allows for the assignment to all items within the population of a prior probability on whether or not it possesses the rare characteristic. This is consistent with the practice of using profiling to select high risk items for inspection. The objective is to find the specific item with a minimal number of inspections. We will determine the optimal inspection strategies for several models according to the average number of inspections needed to find the specific item. Furthermore, an ordering of these models by their average number of inspections is derived. Finally, the use, some discussion, extensions, and examples of the results and conclusions are presented.