Recently the U.S. Senate Committee on Labor and Public Welfare investigated the

Question

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 nonabused 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? What % of children will the screening program identify as abused? (just the 1-in-90 case)

Explanation / Answer

here probability that screening program detect =P(abused and program detect as abused+not abused and program detect as abused)=(1/90)*(0.9)+(1-1/90)*(0.03)=0.039667

hence probability that a child is actually abused given that the screening program makes that diagnosis

=P(abused and program detect as abused)/P(program detect as abused)

=(1/90)*(0.9)/0.039667=0.2521

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