A laboratory blood test is 99 % effective in detecting a certain … - Vidyadip

Question

A laboratory blood test is $99\%$ effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for $0.5\%$ of the healthy person tested $($i.e. if a healthy person is tested, then, with probability $0.005,$ the test will imply he has the disease$).$ If $0.1$ percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?

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Answer

Let $E_{1 }=$ The person selected is suffering from certain disease,
$E_{2 }=$ The person selected is not suffering from certain disease and $A =$ The doctor diagnoses correctly
$\text{Now}\ \ \text{P}(\text{E}_1)=0.1\%=\frac{1}{1000}=0.001,\ \text{P}(\text{E}_2)= 1-\frac{1}{1000}=\frac{999}{1000}=0.999,$
$\text{P}(\text{A}|\text{E}_1)=99\%=\frac{99}{100}=0.99\ \text{P}(\text{A}|\text{E}_2)=0.005\%$
Therefore, by Bayes’ theorem,
$\text{P}(\text{E}_1|\text{A})=\frac{\text{P}(\text{E}_1)\text{P}(\text{A}|\text{E}_1)}{\text{P}(\text{E}_1)\text{P}(\text{A}|\text{E}_1)+\text{P}(\text{E}_2)\text{P}(\text{A}|\text{E}_2)}$
$=\frac{0.01\times0.99}{.001\times0.99+0.999\times0.005}=\frac{990}{990+4995}=\frac{22}{133}$

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