Recently the U.S. Senate Committee on Labor and Public Welfare investigated the feasibility of setting up a national screening program to detect child abuse. A team of consultants estimated the following probabilities:
(1) One child in ninety is abused,
(2) A screening program can detect an abused child 90% of the time, and
(3) A screening program would incorrectly label 3% of all non abused children as abused. What is the probability that a child is actually abused given that the screening program makes that diagnosis? How does the probability change if the incidence of abuse is one in one thousand? Or one in fifty?