Suppose we want to investigate the percentage of abused chil

Suppose we want to investigate the percentage of abused children in a certain population. To do this doctors examine some of these children taken at random from that population. However, doctors are not perfect: They sometimes classify an abused child (A+) as one not abused (D-) or they classify a no abused child (A-) as one that is abused (D+). Suppose these error rates are P(D-|A+)=0.08 and P(D+|A-)=0.05, respectively; thus, P(D+|A+)=0.92 and P(D-|A-)=0.95 are the probabilities of the correct decisions. Let us pretend that only 2% of all children are abused; that is, P(A+)=0.02 and P(A-)=0.98.

A) Select a child at random. What is the probability that the doctor classified this child as abused? That is, compute P(D+)=P(A+)p(D+|A+)+P(A-)P(D+|A-).

B) Compute P(A-|D+) and P(A+|D+).

C) Compute P(A-|D-) and P(A+|D-).

D) Are the probabilities in (b) and (c) alarming? This happens because the error rates of 0.08 and 0.05 are high relative to the fraction 0.02 of abused children in the population.

Solution

A. P(D+)=P(A+)*P(D+|A+)+P(A-)*P(D+|A-)=0.02*0.92+0.98*0.05=0.0674

hence the probability that the doctor classified a randomly selected child as abused is 0.674 [answer]

B. we need to compute these probabilities using BAYES\' THEOREM

P(A-|D+)=P(A-)*P(D+|A-)/(P(A+)*P(D+|A+)+P(A-)*P(D+|A-))=0.98*0.05/0.0674=0.727 [answer]   ,denominator is obtained from part A.

P(A+|D+)=P(A+)*P(D+|A+)/(P(A+)*P(D+|A+)+P(A-)*P(D+|A-))=0.02*0.92/0.0674=0.273 [answer]

C. we need to compute these probabilities using BAYES\' THEOREM

P(A-|D-)=P(A-)*P(D-|A-)/(P(A+)*P(D-|A+)+P(A-)*P(D-|A-))=0.98*0.95/(0.02*0.08+0.98*0.95)=0.98*0.95/0.9326=0.99828 [answer]

P(A+|D-)=(P(A+)*P(D-|A+)/(P(A+)*P(D-|A+)+P(A-)*P(D-|A-))=0.02*0.08/(0.02*0.08+0.98*0.95)=0.0017 [answer]

D. Yes they are alarimng. the probabbility that a child is not abused given that the doctor has also confirmed that is very high..almost equal to one. also the probability that a child is not abused given that the doctor has denied that is also very high.. This happens because the error rates of 0.08 and 0.05 are high relative to the fraction 0.02 of abused children in the population.