A disease effects 1/1000 newborns and shortly after birth a baby is screened for this disease using a cheap test that has a 2% false positive rate (the test has no false negatives). If the baby tests positive, what is the chance it has the disease?I've got P ( disease | positive ) = P ( d p ) P ( p ) = ( 1 / 1000 ) ( 1 / 1000 + 2 / 100 ∗ 999 / 1000 ) = 1 ( 20 + 49 / 50 ) Is this right?

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A disease effects 1/1000 newborns and shortly after birth a baby is screened for this disease using a cheap test that has a 2% false positive rate (the test has no false negatives). If the baby tests positive, what is the chance it has the disease?I&#039;ve got   P  (  disease | positive  )  =            P      (              d        p            )              P      (      p      )        =            (      1              /            1000      )              (      1              /            1000      +      2              /            100      ∗      999              /            1000      )        =      1          (      20      +      49              /            50      )      Is this right?

massetereqe · Accepted Answer

Yes, your answer is correct, but be careful.The reasoning above is correct and works nicely because the false negative rate is   0, so you get that   P  (      d    p    )  =  P  (  d  )  =      1    1000  . Otherwise, you would need to use Bayes&#039; Formula:  P  (  d      |    p  )  =            P      (      p              |            d      )      P      (      d      )              P      (      p      )

Davion Harding · Accepted Answer

This is correct. Alternatively, you could have used Bayes&#039;s theorem:                          P        (        sick                      |                pos        )        =                                        P            (            pos                                  |                        sick            )            ×                      P            (            sick            )                                          P            (            pos                                  |                        sick            )            ×                      P            (            sick            )            +                      P            (            pos                                  |                        healthy            )            ×                      P            (            healthy            )                          .            Now we know that   P  (  sick  )  =  1  /  1000 and   P  (  healthy  )  =  999  /  1000. Also,   P  (  pos    |    sick  )  =  1 (since there are no false negatives: whoever is sick will test positive) and   P  (  pos    |    healthy  )  =  2  /  100 (the rate of false positives).

A disease effects 1/1000 newborns and shortly after birth a baby is screened for this disease using

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