Question

A diagnostic test has the probability $0.95$ of giving a positive result when applied to a person suffering from a certain disease, and a probability $0.10$ of giving a positive result when given to a non-sufferer. It is estimated that $0.5\%$ of the population are suffering from the disease. If this test is now administered to a person from this population about whom there is no information relating to the incidence of this disease and the test gives a positive result, then the probability that he is a sufferer, is

• A. 0.94545
• B. 0.2194
• C. 0.0455
• D. 0.9499

Answer 1

Answer: (C)

Consider the events,
$E_1=$ Person suffering from a certain disease
$E_2=$ Person are not suffering from a certain disease
$A=$ Diagnostic test is positive
$P\left(E_1\right)=0.5\%P\left(E_2\right)=99.5\%$
$P\left(A/E_1\right)=0.95P\left(A/E_2\right)=0.10$
Required probability $P\left(E_1/A\right)=\frac{P\left(E_1\right)\times P\left(A/E_1\right)}{P\left(E_1\right)\times P\left(A/E_1\right)+P\left(E_2\right)\times P\left(A/E_2\right)}$
$=\frac{0.005\times 0.95}{(0.005\times 0.95)+0.995\times 010}$
$=\frac{0.00475}{0.00475+0.0995}=\frac{0.00475}{010425}$
$=\frac{475}{10425}=0.0455$