Question

Suppose that 0.5% 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.

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Answer 1

P[ male professional golfers use steroids ] = 0.5% = 0.005

P[ male professional golfers does not use steroids ] = 1 - P[ male professional golfers use steroids ] =1 - 0.005 = 0.995

P[ test detected that a person takes steroids | male professional golfers does not use steroids ] = 1% = 0.01

P[ test detected that a person does not take steroids | male professional golfers use steroids ] = 10% = 0.1

P[ test detected that a person takes steroids | male professional golfers use steroids ] = 1 - P[ test detected that a person does not take steroids | male professional golfers use steroids ] = 1 - 0.1 = 0.9

Max actually uses steroids if he tests positive for steroid use = P[ Max actually uses steroids | test detected that a Max takes steroids ]

P[ Max actually uses steroids | test detected that a Max takes steroids ] = P[ Max actually uses steroids and test detected that a Max takes steroids ] / P[ test detected that a Max takes steroids ]

P[ test detected that a Max takes steroids ] = P[ Max actually uses steroids and test detected that a Max takes steroids ] + P[ Max actually does not use steroids and test detected that a Max takes steroids ]

P[ Max actually uses steroids and test detected that a Max takes steroids ] = P[ test detected that a person takes steroids | male professional golfers use steroids ]*P[ male professional golfers use steroids ] = 0.9*0.005 = 0.0045

P[ Max actually does not use steroids and test detected that a Max takes steroids ] = P[ test detected that a person takes steroids | male professional golfers does not use steroids ]*P[ male professional golfers does not use steroids ] = 0.01*0.995 = 0.00995

P[ test detected that a Max takes steroids ] = 0.0045 + 0.00995 = 0.01445

P[ Max actually uses steroids | test detected that a Max takes steroids ] = 0.0045/ 0.01445

P[ Max actually uses steroids | test detected that a Max takes steroids ] = 0.3114