A manufacturer claims that its drug test will detect steroid use (that is, show positive for an athlete who uses steroids) 95% of the time. What the company does not tell you is that 15% of all steroid-free individuals also test positive ( i.e. the false positive rate). 10% of the football team members use steroids. Your friend on the football team has just tested positive, what is the probability that he uses steroids?

#### Solution

Let $A$ denote positive test result and $B$ denote uses steriods.

Given that 10% of the football team members use steroids, i.e., $P(B) = 0.1$. Thus $P(B^\prime) = 1-P(B)=0.90$.

A manufacturer claims that its drug test show positive for an athlete who uses steroids) 95% of the time i.e., $P(A|B) = 0.95$.

15% of all steroid-free individuals also test positive i.e., $P(A|B^\prime)= 0.15$.

Using Bayes' Theorem, required probability is

` $$ \begin{aligned} P(B|A) &=\frac{P(A|B)P(B)}{P(A)}\\ &= \frac{P(A|B)P(B)}{P(A|B)P(B)+P(A|B^\prime)P(B^\prime)}\\ &=\frac{0.95\times 0.1}{0.95\times 0.1 + 0.15\times 0.90}\\ &=\frac{0.095}{0.23}\\ &=0.413 \end{aligned} $$ `

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators