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. Further, 15% of all steroid-free
Individuals also test positive. 10% of the rugby team members use steroids. Your friend
On the rugby team has just tested positive. The probability that he uses steroids is
A) 0.4130
B) 0.8636
C) 0.0950
D) 0.2300

Question

Quizplus · Accepted Answer

To find the probability that your friend uses steroids given that he tested positive, we can use Bayes' theorem. Let's denote:- $P(S)$ as the probability of using steroids, which is 0.10.- $P(\neg S)$ as the probability of not using steroids, which is 0.90.- $P(+|S)$ as the probability of testing positive given steroid use, which is 0.95.- $P(+|\neg S)$ as the probability of testing positive without steroid use, which is 0.15.We want to find $P(S|+)$, the probability of using steroids given a positive test. Bayes' theorem gives us:$$P(S|+) = \frac{P(+|S) \cdot P(S)}{P(+|S) \cdot P(S) + P(+|\neg S) \cdot P(\neg S)}$$Plugging in the numbers:$$P(S|+) = \frac{0.95 \cdot 0.10}{0.95 \cdot 0.10 + 0.15 \cdot 0.90} = \frac{0.095}{0.095 + 0.135} = \frac{0.095}{0.23} = 0.4130$$Therefore, the probability that your friend uses steroids given that he tested positive is 0.4130.

A Manufacturer Claims That Its Drug Test Will Detect Steroid