In a certain village, 20% of the population has some disease. A test is administered which has the property that if a person is sick, the test will be positive 90% of the time and if the person is not sick, then the test will still be positive 30% of the time. All people tested positive are prescribed a drug which always cures the disease but produces a rash 25% of the time. Given that a random person has the rash, what is the probability that this person had the disease to start with?I am looking for P(S|R) given that a person tested positive where S denotes a sick person R denotes a person with a rash, given that they tested positive. If + denotes a person who tested positive and I use Baye's formula and the data to calculate P ( S | + ) would P ( S | R ) = P ( R | + ) P ( S | + ) P ( R | + ) = P ( S | + )? Or would the answer be P ( S | R ) = P ( R ) P ( S | + )? Or are both of these answers wrong? Also, I cannot tell if in the problem statement P ( R ) = .25 or P ( R | + ) = .25

Question

In a certain village, 20% of the population has some disease. A test is administered which has the property that if a person is sick, the test will be positive 90% of the time and if the person is not sick, then the test will still be positive 30% of the time. All people tested positive are prescribed a drug which always cures the disease but produces a rash 25% of the time. Given that a random person has the rash, what is the probability that this person had the disease to start with?I am looking for P(S|R) given that a person tested positive where S denotes a sick person R denotes a person with a rash, given that they tested positive. If + denotes a person who tested positive and I use Baye&#039;s formula and the data to calculate   P  (  S      |    +  ) would   P  (  S      |    R  )  =            P      (      R              |            +      )      P      (      S              |            +      )              P      (      R              |            +      )        =  P  (  S      |    +  )? Or would the answer be   P  (  S      |    R  )  =  P  (  R  )  P  (  S      |    +  )? Or are both of these answers wrong? Also, I cannot tell if in the problem statement   P  (  R  )  =  .25 or   P  (  R      |    +  )  =  .25

Gilbert Petty · Accepted Answer

Step 1Assuming that the rash occurs only due to the medicine and not for other reasons,First, P(S)=0.2,P(Sc)=0.8What is the probability that a sick person tests positive and finally develops rashes after taking medicine? P(R|S)=0.9×0.25Step 2Similarly, find P(R|Sc)Now      P    (    S          |        R    )    =                                                                  P            (            R                          |                        S            )            ⋅            P            (            S            )                                                                                      P            (            R            )                                    =                                                                  P            (            R                          |                        S            )            ⋅            P            (            S            )                                                                                      P            (            R                          |                        S            )            ⋅            P            (            S            )            +            P            (            R                          |                                      S              c                        )            ⋅            P            (                          S              c                        )

Emmanuel Giles · Accepted Answer

Step 1We want  P  (  S      |    R  )  =            P      (      R              |            S      )      P      (      S      )      )              P      (      R      )      We are given P(S),   P  (  +      |    S  ) and   P  (  +      |    ¬  S  ) and, indirectly,   P  (  R      |    +  ). I say indirectly because we are actually given the probability that a person gets a rash given that he is administered the drug but this only happens when a person tests positive.We must use the given probabilities and P(+) to calculate the unknowns P(R|S) and P(R).  P  (  +  )  =  P  (  +      |    S  )  P  (  S  )  +  P  (  +      |    ¬  S  )  P  (  ¬  S  )  =  0.9  ∗  0.2  +  0.3  ∗  0.8  =  0.42    P  (  R  )  =  P  (  R      |    +  )  P  (  +  )  +  P  (  R      |    ¬  +  )  P  (  ¬  +  )  =  0.25  ∗  0.42  +  0  ∗  0.58  =  0.105    P  (  R      |    S  )  =  P  (  R      |    +  )  P  (  +      |    S  )  =  0.25  ∗  0.9  =  0.225Plugging in the values above gives   P  (  S      |    R  )  =            0.225      ∗      0.2        0.105    =      3    7

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