Question

Suppose that 0.7% of male professional golfers use steroids, and that Max is a male professional golfer who has been randomly selected to take a drug test. The test he has been asked to take has a false positive rate of 1% and a false negative rate of 10%. Use Bayes’ rule to calculate the probability that Max actually uses steroids if he tests positive for steroid use. Give your answer as a decimal precise to three decimal places.

Answer 1

Answer: 0.388

Step-by-step explanation:

According to Bayes' theorem:

$P(steroids\:|\:positive) = \frac{P(positive\:|\:steroids)\times P(steroids)}{P(positive)}$

Whereby:

The probability of positive result given the use of steroid:

$P(positive\:|\:steroids) = 1 - P(false\:negative) = 1 - 0.1 = 0.9$

The probability of Max using steroids: $P(steroids) = 0.7\% = 0.007$

The probability of positive result (calculated using addition and multiplication rules):

$P(positive)= P(positive\:\cap \:steroids) + P(positive\:\cap \:non-steroids)\\ =P(steroids)\times P(positive\:|\:steroids) + P(non-steroids)\times P(positive|non-steroids)\\= 0.007\times0.9+(1-0.007)\times 0.01 = 0.01623$

As such,

$P(steroids\:|\:positive) = \frac{0.9\times 0.007}{0.01623} \approx 0.388$