Kathy Guerra

Kathy Guerra

Answered

2022-09-08

A population includes a fraction m of individuals carrying a disease exists and has the following characteristics:
P(positive test | individual with the disease) = p
P(positive test | individual without the disease) = r
What is the probability of a false negative: P(individual with the disease | negative test)?
My try
Let W = with the disease, ~W = without the disease;
Let + = positive test, - = negative test
From the given, we have: P(W) = m, P(~W) = 1-m
We 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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Answer & Explanation

Carson Mueller

Carson Mueller

Expert

2022-09-09Added 7 answers

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 ).

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elisegayezm

elisegayezm

Expert

2022-09-10Added 1 answers

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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