5. A laboratory test for steroid use in professional athletes has detection rate

Question

5. A laboratory test for steroid use in professional athletes has detection rates given in the following table: Test Result Steroid Use Yes No 0.90 0.10 0.01 0.99 (a) A randomly selected professional athlete tests positive for steroid use. If the rate of steroid use among professional athletes is 1 in 100, what is the probability that the athlete has actually been using steroids? (b) If the rate of steroid use among professional athletes is 55% (the percentage in major league baseball, at least according to media reports and former players), what is the probability that the athlete has actually been using steroids given that he had a positive test? (c) Do you think Major League Baseball would be justified in suspending a player based on a single positive drug test? Explain your decision.

Explanation / Answer

a)P(tested positive)=P(using steroids and tested posiive)+P(not using steroids and tested posiive)

=(1/100)*0.9+(1-1/100)*0.01=0.0189

P(using steroids|tested posiive)=P(using steroids and tested posiive)/P(tested positive)

=(1/100)*0.9/0.0189=0.47619

b)P(tested positive)=P(using steroids and tested posiive)+P(not using steroids and tested posiive)

=0.55*0.9+(1-0.55)*0.01=0.4995

P(using steroids|tested posiive)=P(using steroids and tested posiive)/P(tested positive)

=0.55*0.9/0.4995=0.9910

c)

as probabiltiy from part b) is very high for a person to use steriod given tested postiive ; therefore Major league Baseball is justified in suspending a player based on a single posiitve drug test on part b)

but not in part a)

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