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)?
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)
Answer & Explanation
You have it almost right. You write , and you know everything except and .
On the other hand, you write the correct expression for , but then you know . That is, .
Compute directly instead of .
The probability that the test is negative is
The probability that the test is negative and the person has the disease is
Therefore, the probability that a person has the disease, given that the test is negative, is
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