A population includes a fraction m of individuals carrying a disease exists and has the following characteristics:P(positive test | individual with the disease) = pP(positive test | individual without the disease) = rWhat is the probability of a false negative: P(individual with the disease | negative test)?My tryLet W = with the disease, ~W = without the disease;Let + = positive test, - = negative testFrom the given, we have: P(W) = m, P(~W) = 1-mWe want P(W | -) = (P(- |W) P(W))/(P(-)) = (P(- | W)P(W)/(P(- | W)P(W) + P(- ~W)P(~W))We have: P (+) = P(+|W)P(W) + P(+|~W)P(~W) = p * P(W) + r * P(~W)

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Carson Mueller · Accepted Answer

You have it almost right. You write   P  (  W      |    −  )  =  P  (  −      |    W  )  P  (  W  )      /    P  (  −  ), and you know everything except   P  (  −      |    W  )  =  1  −  p and   P  (  W  )  =  m.On the other hand, you write the correct expression for   P  (  +  ), but then you know   P  (  −  )  =  1  −  P  (  +  ). That is,   P  (  −  )  =  1  −  p  m  −  r  (  1  −  m  ).

elisegayezm · Accepted Answer

Alternate solution:Compute directly   P  (  −  ) instead of   P  (  +  ).The probability that the test is negative is  P  (  −  )  =  P  (  W  )  P  (  −      |    W  )  +  P  (  ∼  W  )  P  (  −      |    ∼  W  )  =  m  (  1  −  p  )  +  (  1  −  m  )  (  1  −  r  )The probability that the test is negative and the person has the disease is  P  (  −  ∩  W  )  =  P  (  W  )  P  (  −      |    W  )  =  m  (  1  −  p  )Therefore, the probability that a person has the disease, given that the test is negative, is  P  (  W      |    −  )  =            P      (      −      ∩      W      )              P      (      −      )        =            m      (      1      −      p      )              m      (      1      −      p      )      +      (      1      −      m      )      (      1      −      r      )

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