A hospital identifies patients with an average false positive rate of 0.01. If it checks 31333 patients and returns a total of 357 positive results, then what's the expected number of true positives?

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A hospital identifies patients with an average false positive rate of 0.01. If it checks 31333 patients and returns a total of 357 positive results, then what&#039;s the expected number of true positives?

atoandro8f04v · Accepted Answer

Let   С represent the event that a randomly selected patient has a condition that the test is intended to detect. Let   T represent the event that a randomly selected patient tests positive for the condition. Then a false positive is the event   T  ∣      C    ¯  , i.e., a patient without the condition, when tested, yields a positive result.You are given  Pr  [  T  ]  =      357    31333    ,and  Pr  [  T  ∣      C    ¯    ]  =  0.01.You want to compute  Pr  [  T  ∣  C  ]  ,the probability of a true positive. To this end, we write  Pr  [  T  ]  =  Pr  [  T  ∣  C  ]  Pr  [  C  ]  +  Pr  [  T  ∣      C    ¯    ]  Pr  [      C    ¯    ]  .Since   Pr  [  C  ]  +  Pr  [      C    ¯    ]  =  1, we have  Pr  [  T  ∣  C  ]  =            Pr      [      T      ]      −      Pr      [      T      ∣                  C        ¯            ]      (      1      −      Pr      [      C      ]      )              Pr      [      C      ]        =  0.00139374  +      1          100      Pr      [      C      ]        .Without knowing the underlying prevalence of the condition, we see that it is not possible to uniquely determine the desired conditional probability, and therefore we also cannot determine the expected number of true positives.

A hospital identifies patients with an average false positive rate

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