Question

an x-ray test is used to detect a certain disease that is common in 3% of the population the test has the following error 87 per-cent of people who are disease-free to get a positive reaction and 2% of the people who have the disease do get a negative reaction from the test a large number of people are screened at random using the test and those with a positive reaction are further examine what is the probability that a person tested at random indeed has the disease given that the test results shows positive

Answer 1

Let the event D be that a person has the disease.

$P(D)=\frac{3}{100},P(D^{\prime})=\frac{97}{100}$

Now Let T be positive test and T' be a negative test result.

$P(T|D^{\prime})=\frac{87}{100},P(T|D)=\frac{13}{100},P(T^{\prime}|D)=\frac{2}{100},P(T^{\prime}|D^{\prime})=\frac{98}{100}$

So the probability that the person has the disease given that the test is positive is given by:

$\begin{gathered} P(D|T)=\frac{P(D)P(T|D)}{P(D)P(T|D)+P(D^{\prime})P(T|D^{\prime})} \\ =\frac{\frac{3}{100}\times\frac{13}{100}}{\frac{3}{100}\times\frac{13}{100}+\frac{97}{100}\times\frac{87}{100}} \\ =\frac{13}{2826} \\ \approx0.0046 \end{gathered}$

So only 0.46% of the people who tested positive will have the disease.