Question

Use the following results from a test for marijuana​ use, which is provided by a certain drug testing company. Among 149 subjects with positive test​ results, there are 21 false-positive​ results; among 158 negative​ results, there are 2 false-negative results. If one of the test subjects is randomly​ selected, find the probability that the subject tested negative or did not use marijuana.​

The probability that a randomly selected subject tested negative or did not use marijuana is

Answer 1

P(Not use marijuana | tested Positive) = 21 / 149 = 0.1409396

P(Use marijuana | tested Negative) = 2 / 158 = 0.01265823

=> P(Not use marijuana | tested Negative) = 1 - P(Use marijuana | tested Negative) = 1- 0.01265823 =  0.9873418

P(Tested Negative) = Total negative results / Total positive and negative results =  158 / (149 + 158) = 0.514658

P(Tested Positive) = 1 - P(Tested Negative) = 1 - 0.514658 = 0.485342

By law of total probability,

P(Not use marijuana) =  P(Not use marijuana | tested Positive) P(Tested Positive) + P(Not use marijuana | tested Negative)  P(Tested Negative)

= 0.1409396 * 0.485342 + 0.9873418 * 0.514658

= 0.5765473

P(Not use marijuana or tested Negative) = P(Not use marijuana) + P(Tested Negative) - P(Not use marijuana and tested Negative)

= P(Not use marijuana) + P(Tested Negative) - P(Not use marijuana | tested Negative) P(Tested Negative)

= 0.5765473 + 0.514658 - 0.9873418 * 0.514658

= 0.5830619