A hospital identifies patients with an average false positive rate

Schenone2pare

Schenone2pare

Answered question

2022-06-02

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?

Answer & Explanation

atoandro8f04v

atoandro8f04v

Beginner2022-06-03Added 7 answers

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.

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