Search any question & find its solution
Question:
Answered & Verified by Expert
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
Options:
Solution:
1799 Upvotes
Verified Answer
The correct answer is:
0.0455
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
$\begin{aligned}
& P\left(E_1\right)=0.5 \% \quad P\left(E_2\right)=99.5 \% \\
& P\left(A / E_1\right)=0.95 P\left(A / E_2\right)=0.10 \\
&
\end{aligned}$
Required probability
$\begin{aligned}
& P\left(E_1 / A\right)=\frac{P\left(E_1\right) \times P\left(A / E_1\right)}{P\left(E_1\right) \times P\left(A / E_1\right)+P\left(E_2\right) \times P\left(A / E_2\right)} \\
& =\frac{0.005 \times 0.95}{(0.005 \times 0.95)+0.995 \times 0.10} \\
& \quad=\frac{0.00475}{0.00475+0.0995}=\frac{0.00475}{0.10425}=\frac{475}{10425}=0.0455
\end{aligned}$
$E_1=$ Person suffering from a certain disease
$E_2=$ Person are not suffering from a certain disease
$A=$ Diagnostic test is positive
$\begin{aligned}
& P\left(E_1\right)=0.5 \% \quad P\left(E_2\right)=99.5 \% \\
& P\left(A / E_1\right)=0.95 P\left(A / E_2\right)=0.10 \\
&
\end{aligned}$
Required probability
$\begin{aligned}
& P\left(E_1 / A\right)=\frac{P\left(E_1\right) \times P\left(A / E_1\right)}{P\left(E_1\right) \times P\left(A / E_1\right)+P\left(E_2\right) \times P\left(A / E_2\right)} \\
& =\frac{0.005 \times 0.95}{(0.005 \times 0.95)+0.995 \times 0.10} \\
& \quad=\frac{0.00475}{0.00475+0.0995}=\frac{0.00475}{0.10425}=\frac{475}{10425}=0.0455
\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.