Answer:
The probability is  [tex]P(U | P ) = 11.6 \%[/tex]
Step-by-step explanation:
Generally a true positive means that the person tested is a drug user and he/she test positive to the test
while true negative mean that the person tested is not a drug user and he/she test negative to the test
From the question we are told that
The probability that a person test positive  to the drug test given that the person is  a drug user is  Â
   [tex]P(P | U ) = 0.93[/tex]
Here  P => event that a person test positive to the drug test
     U => event that the person is a drug user
The probability that a person test negative to the drug test given that the person is not a drug user is   Â
   [tex]P(N | S ) = 0.87[/tex] Â
Here  N => event that a person test negative to the drug test
     S => event that the person is not a drug user
The probability that a person is a drug user is Â
   [tex]P(U) = 0.018[/tex]
Generally the probability that a person test positive to the test given that the person is a non- user is Â
   [tex]P(P | S ) = 1 - 0.87[/tex]
=> Â [tex]P(P | S ) = 0.13[/tex]
Generally the probability that a person is  a non user of drug is Â
   [tex]P(S) = 1 - 0.018[/tex] Â
=> [tex]P(S) = 0.982[/tex] Â
Generally the probability that a person test positive to the drug test is mathematically evaluated as
   [tex]P(P) = P(P | U) * P(U ) + P(P | S ) * P(S)[/tex]
=> Â [tex]P(P) = 0.93 * 0.018 + 0.13 * 0.982[/tex]
=> Â [tex]P(P) = 0.1444[/tex]
Generally the probability a person is a user given that he or she tested positive is mathematically represented as
   [tex]P(U | P ) = \frac{P(P |U ) * P(U)}{P(P)}[/tex]
=> Â [tex]P(U | P ) = \frac{0.93 * 0.018 }{0.1444}[/tex]
=> Â [tex]P(U | P ) = 0.116[/tex]
Converting to percentage
  [tex]P(U | P ) = 0.116 * 100[/tex]
=> [tex]P(U | P ) = 11.6 \%[/tex]
