Question

Drug testing in athletes.When Olympic athletes are tested for illegal drug use (i.e., doping), the results of a single positive test are used to ban the athlete from competition. Chance(Spring 2004) demonstrated the application of Bayes’s Rule for making inferences about testosterone abuse among Olympic athletes using the following example: In a population of 1,000 athletes, suppose 100 are illegally using testosterone. Of the users, suppose 50 would test positive for testosterone. Of the nonusers, suppose 9 would test positive.

1. Given that the athlete is a user, find the probability that a drug test for testosterone will yield a positive result. (This probability represents the sensitivity of the drug test.)
2. Given the athlete is a nonuser, find the probability that a drug test for testosterone will yield a negative result. (This probability represents the specificityof the drug test.)
3. If an athlete tests positive for testosterone, use Bayes’s Rule to find the probability that the athlete is really doping. (This probability represents the positive predictive value of the drug test.)

Short answer

1. The probability that a drug test for testosterone will yield a positive result is 0.5.
2. The probability that a drug test for testosterone will yield a negative result Is 0.09.
3. The probability that the athlete is really doping is 0.8475.

Step-by-step solution

Important formula

The formula for probability is $\mathbf{P}\mathbf{=}\frac{\mathrm{Favourable}\mathrm{out}\mathrm{comes}}{\mathrm{Total}\mathrm{out}\mathrm{comes}}$

(a) Find the probability that a drug test for testosterone will yield a positive result

The sample is:

A = athletes use the substances.

H = healthy subject is test positive

$\begin{array}{c}\mathbf{P}\mathbf{\left(}\mathbf{H}\mathbf{\right|}\mathbf{A}\mathbf{)}\mathbf{=}\frac{50}{100}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{5}\end{array}$

So, the probability that a drug test for testosterone will yield a positive result is 0.5.

(b) Determine the probability that a drug test for testosterone will yield a negative result

$\begin{array}{l}\mathbf{P}\mathbf{\left(}\mathbf{H}\mathbf{\right|}\mathbf{A}\mathbf{)}\mathbf{=}\frac{\mathbf{no}\mathbf{.}\mathbf{}\mathbf{of}\mathbf{}\mathbf{athlete}\mathbf{\text{'}}\mathbf{s}\mathbf{}\mathbf{negative}\mathbf{}\mathbf{in}\mathbf{}\mathbf{substance}}{\mathbf{total}\mathbf{}\mathbf{no}\mathbf{.}\mathbf{}\mathbf{of}\mathbf{}\mathbf{persons}\mathbf{}\mathbf{use}\mathbf{}\mathbf{illeagal}\mathbf{}\mathbf{substance}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{9}{100}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{09}\end{array}$

Hence, the probability that a drug test for testosterone will yield a negative result Is 0.09.

(c) Find the probability that the athlete is really doping

By using the Bayes’ rule.

$\begin{array}{l}\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right|}\mathbf{H}\mathbf{)}\mathbf{=}\frac{\mathbf{P}\mathbf{\left(}\mathbf{H}\mathbf{\right|}\mathbf{A}\mathbf{)}\mathbf{.}\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}}{\mathbf{P}\mathbf{\left(}\mathbf{H}\mathbf{\right|}\mathbf{A}\mathbf{)}\mathbf{.}\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}\mathbf{H}\mathbf{\right|}{\mathbf{A}}^C\mathbf{\left)}\mathbf{.}\mathbf{P}\mathbf{(}{\mathbf{A}}^C\mathbf{\right)}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{5}\mathbf{\left)}\mathbf{\right(}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{)}}{\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{5}\mathbf{\left)}\mathbf{\right(}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{)}\mathbf{+}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{01}\mathbf{\left)}\mathbf{\right(}\mathbf{0}\mathbf{.}\mathbf{9}\mathbf{)}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{8475}\end{array}$

Therefore, the probability that the athlete is really doping is 0.8475.