A person who does not take the drugs will test positive with probability 0.02

Question

A sports association decides to implement a drug screening procedure to test its athletes for illegal performance enhancing drugs. A person who does not take the drugs will test positive with probability 0.02 and a person who does take the drugs will test negative with probability 0.04. Suppose that 3% of the athletes tested take performance enhancing drugs. What is the probability that…

a. an athlete testing positive takes the drugs?

b. an athlete testing positive does not take the drugs?

c. an athlete testing negative takes the drugs?

d. an athlete testing negative does not take the drugs?

I was given a hint that I can use Bayes Theorem, but I have no idea how to do this and it looks complicated.. Can anyone help me to solve this..

Answer 1

Refer to the probability tree diagram (D-take drug, D'-do not take drug, P-positive, N-negative):

a) $P(D|P)=\frac{P(D)P(P|D)}{P(D)P(P|D)+P(D')P(P|D')}=\frac{0.0288}{0.0288+0.0194}=0.5975.$

b) $P(D'|P)=\frac{P(D')P(P|D')}{P(D')P(P|D')+P(D)P(P|D)}=\frac{0.0194}{0.0194+0.0288}=0.4025.$

c) $P(D|N)=\frac{P(D)P(N|D)}{P(D)P(N|D)+P(D')P(N|D')}=\frac{0.0012}{0.0012+0.9506}=0.0013.$

d) $P(D'|N)=\frac{P(D')P(N|D')}{P(D')P(N|D')+P(D)P(N|D)}=\frac{0.9506}{0.9506+0.0012}=0.9987.$

Answer 2

If you are someone looking at these kind of probabilities for the first time, it helps to first convert the data into percentages of the whole population of athletes.

Test Positive/Take Drugs 2.88% (96% of 3%)

Test Positive/No Drugs 1.94% (2% of 97%)

Test Negative/Take Drugs 0.12% (4% of 3%)

Test Negative/No Drugs 95.06% (98% of 97%)

Answer to a. An athlete testing positive takes drugs.......

The conditional part here is, given someone tests positive what portion actually takes drugs. So the percentage that test positive is 2.88 + 1.94 = 4.82%. Of these 2.88% actually take drugs so the probability is 2.88/4.82 = .598

In proper notation $P(A/B) = P(B/A)P(A)/P(B)$

Where A is taking drugs and B is testing positive

$= \frac{.96*.03}{(.96*.03)+(.97*.02)} = \frac{.0288}{.0482} = .598$