Question

2. In recent years, the “Tour de France” cycling race has been plagued with accusations that cyclists have taken steroids to boost their performance. Testing for these steroids has been difficult. The level of steroids that naturally occur in the human body can vary between individuals and the levels can change throughout the day. Testing for the presence of steroids is expensive and time- consuming. Racers are randomly chosen for drug tests.

Suppose that 2% of racers have taken an illegal steroid. Assume that if they took the drug, there is a 99% chance that the test will return a “positive” reading and the athlete will be disqualified. These tests are not perfect. Even if the cyclist did not take a steroid, there is still a 3% chance that the test

will return a “positive” reading and the athlete will be unjustly disqualified.

a) Draw a tree diagram showing the probabilities for the sequence of events in which an athlete either takes or does not take a drug, and then is tested for the presence of the drug.

b) What is the probability that a randomly chosen cyclist will be steroid-free and will still test positive for steroids?

c) What is the probability that a randomly chosen cyclist will be disqualified due to a “positive” reading?

Answer 1

(a)

D = event that someone has taken drug.

P(D) = 0.02

C = event that someone is caught and disqualified

P( D C ) = 0.02 * 0.99 = 0.0198

P( D' C ) = 0.98 * 0.03 = 0.0294

P( D C' ) = 0.02 * 0.01 = 0.002

P( D' C' ) = 0.98 * 0.97 = 0.9506

(b)

P( randomly chosen cyclist will be steroid free and will still test for steroids )

= P( C D' ) =0.0294

(c)

P( randomly chosen cyclist will be disquilified due to positive reading )

= P( C D ) + P( C D')

= 0.0198 + 0.0294

= 0.0492