Question

It is estimated that 3% of the athletes competing in a large tournament are users of an illegal drug to enhance performance. Athletes are tested to see if they are using illegal drugs or not using illegal drugs and the test can come back positive or negative. The test for this drug is 90% accurate for non-drug users (this means there is a 90% chance the test comes back negative if the person is a non-user). What is the probability that an athlete tests negative and is not a user of an illegal drug

Answer 1

0.873 = 87.3% probability that an athlete tests negative and is not a user of an illegal drug

Explanation:

Conditional Probability

We use the conditional probability formula to solve this question. It is

`P(B|A) = (P(A \cap B))/(P(A))`

In which

P(B|A) is the probability of event B happening, given that A happened.

`P(A \cap B)` is the probability of both A and B happening.

P(A) is the probability of A happening.

In this question:

Event A: Not a user of an illegal drug.

Event B: Tests negative.

The test for this drug is 90% accurate for non-drug users (this means there is a 90% chance the test comes back negative if the person is a non-user)

This means that
`P(B|A) = 0.9`

It is estimated that 3% of the athletes competing in a large tournament are users of an illegal drug to enhance performance.

So 100 - 3 = 97% probability that the athlete does not use the drug, so
`P(A) = 0.97`

What is the probability that an athlete tests negative and is not a user of an illegal drug

`P(B|A) = (P(A \cap B))/(P(A))`

`P(A \cap B) = 0.9*0.97 = 0.873`

0.873 = 87.3% probability that an athlete tests negative and is not a user of an illegal drug