What is the probability that an athlete is doping with testosterone
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What is the probability that an athlete is doping with testosterone

In a population of 1,000 athletes, suppose 100 are doping with testosterone. Of the users, suppose 50 would test positive for testosterone. Of the nonusers, suppose 9 would test positive. 

Let U be the event that an athlete is a testosterone user and therefore Uc is the event that an athlete is not a testosterone user. Let N be the event that an athlete tests negative for testosterone and therefore Nc is the event that an athlete tests positive for testosterone.
(a) What is the probability that an athlete is doping with testosterone?
(b) What is the probability that an athlete is doping with testosterone and tests positive?
(c) Given the athlete is a testosterone user, find the probability that a drug test for testosterone will yield a positive result. (This probability represents the sensitivity of the drug test.)
(d) Given the athlete is a nonuser, find the probability that a drug test for testosterone will yield a negative result. (This probability represents the specificity of the drug test.)
(e) Construct a tree diagram for the above scenario. Clearly label the arms representing sensitivity and specificity.
(f) Given that an athlete tests positive for testosterone find the probability that the athlete is really doping. (This probability represents the positive predictive value of the drug test.)

Hint

Let U be the event that an athlete is a testosterone user
Uc is the event that an athlete is not a testosterone user
N be the event that an athlete tests negative for testosterone
Nc is the event that an athlete tests positive for testosterone

Testing

Drug

Positive (Nc)

Negative (N)

Total

User (U)

50

50

100

Non user (Uc)

9

891

900

total

59

941

1000

Hint
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