Question

1. A new test is developed for detecting the presence of steroids in athlete urine samples. The test could produce false negative results 5% of the time. It could also produce false positive results 10% of the time. Based on past evidence, it is believed that 2% of the athletes do use steroids.

1. This test is administered to an athlete, and his test result is positive for steroids. What is the probability that this athlete actually used steroids? Show your work. [2 points]
2. An athlete who does not use steroids was tested positive for steroid use. In response to her complaint, the athletics officials agreed to conduct her steroid test again. What is the probability that her test will turn positive again? Show/explain how you arrived at your work.    
3. What is the probability that a steroid user tests negative twice in a row? Show/explain your work.     

Answer 1

a)

P(tested positive) =P(use)*P(tested positive |use)+P(not use)*P(tested positive |not use)

=0.02*(1-0.05)+(1-0.02)*0.1 =0.117

therefore P(use |tested positive) =P(use)*P(tested positive |use)/P(tested positive)

=0.02*(1-0.05)/0.117 =0.1624

b)since test are assumed to be independent from trail to trail :

P(turn positive again |not use steroid) =P(false positive) =0.10

c)since again events are independent from trail to trail

P(tested negative twice given |uses) =P(false negative on first trail)*P(false negative on 2nd trial) =0.05*0.05 =0.0025