A particular test for the presence of steroids is to be used after a professiona

Question

A particular test for the presence of steroids is to be used after a professional tract meet. If steroids are present, the test will accurately indicate this 95% of the time. However, if steroids are not present, the test will indicate this 90% of the time (so it is wrong 10% of the time and predicts the presence of steroids). Based on past data, it is believed that 2% of the athletes do use steroids. This test is administered to one athlete, and the test is positive for steroids. What is the probability that this person actually used steroids?

Explanation / Answer

Given:

P( test +ve|Sterioid used) =95% = 0.95

P( Test -ve | Steroid not used ) = 90% =0.9;P( Test +ve | Steroid not used )=1-0.9=0.1

P( Steroid used ) =2% =0.02; P( Steroid not used )=1-0.02 =0.98

P(Steroid used | Test +ve) = P( 'Steroid used' And 'Test +ve')/P(Test +ve)

P(test +ve ) = P( Steroid used )*P( test +ve|Sterioid used) + P( Steroid not used )*P( Test +ve | Steroid not used )

=(0.02*0.95) + (0.98*0.1) =0.117

P( 'Steroid used' And 'Test +ve')=P( Steroid used )*P( test +ve|Sterioid used)=(0.02*0.95)=0.019

P(Steroid used | Test +ve) = P( 'Steroid used' And 'Test +ve')/P(Test +ve) =0.019/0.117= 0.1624

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