Question

Let’s say that we work for the International Olympic Committee (IOC) as part of their Fight Against Doping. We have a drug test for a banned performance-enhancing drug (PED) that is 99.3% accurate at identifying the presence of the PED in an athlete’s system. However, it is only 73% accurate at identifying the absence of PED in the athlete’s system. From a scientific study we also have a strong a priori reason to believe that only 3% of Olympic athletes use this particular PED. Use this information, along with Bayes’ Theorem, to answer the following questions:

1. An athlete tests positive for the PED. Given this positive result, what is the probability the tested individual uses the PED?

2. As an employee of the IOC, we don’t want to needlessly ban an athlete from the chance to compete in the Olympics. As a result, we decide to institute a protocol that if an athlete tests positive for the use of the PED we will administer a second test. The second test is less accurate at identifying an athlete that has the PED in their system, at only 81%, but is more accurate at identifying the absence of PED in the athlete’s system, with a probability of 90%. If the athlete tests positive for the PED in both the first and second test, what is the probability that the accused individual uses the banned PED? You may assume the outcome of the second drug test is conditionally independent of the outcome of the first drug test. Hint: Think about Lesson 6, and how to calculate the probability of two independent events both occurring.

3. Our information that only 3% of Olympic athletes use the PED came from a study of 300 athletes. This year we tested 500 athletes and confirmed that 11 of them used the banned substance. In both cases, only a sample of all athletes to complete in the Olympics were tested for the PED. As a result there is some uncertainty, so we decide we would like to express what we have learned as a beta distribution. In two years when we being testing athletes again, what is our updated prior distribution for the percentage of Olympic athletes using the PED?

Answer 1