Question

A certain disease has an incidence rate of 0.1%. If the false-negative rate is 6% and the false positive rate is 1%, compute the probability that a person who tests positive actually has the disease. ____ Give your answer accurate to at least 3 decimal places.

Answer 1

Step-by-step explanation

To solve this problem, we must apply Bayes Theorem, which states that:

`P(A|B)=\fracA){P(A)P(B|A)+P(\bar{A})P(\bar{B}|\bar{A)}}.`

We define the events:

• A = has the disease,

,

• B = test positive.

From the statement, we know that:

• the disease has an incidence rate of 0.1% → P(A) = 0.1% = 0.001 → P(not A) = 99.9% = 0.999,

• anyone who has the disease will test positive → P(B | A) = 100% = 1,

• the false positive rate is 1% → P(not A) = 1% = 0.01,

,

• the false-negative rate is 6% → P(not B | not A) = 6% = 0.06.

Replacing these values in the formula above, we get:

`P(A|B)=(0.001*1)/(0.001*1+0.999*0.06)\cong0.016.`Answer

The probability that a person who tests positive actually has the disease is approximately 0.016.