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 diseases). 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?
.: Let ${E_1}:$ 'the person has the disease' and ${E_2}:$ 'the person is healthy',
$\Rightarrow$ $P\left( {{E_1}} \right) = 0.1\% = \cfrac{{0.1}}{{100}} = \cfrac{1}{{1000}} = 0.001$
and $P\left( {{E_2}} \right) = 1 - \cfrac{1}{{1000}} = \cfrac{{999}}{{1000}} = 0.999$
Let $A$ : 'test is positive', then
$P\left( {A|{E_1}} \right) = \cfrac{{99}}{{100}} = 0.99{\rm{ and }}P\left( {A|{E_2}} \right) = 0.005$
Hence, the required probability is
$P\left( {{E_1}|A} \right) = \cfrac{{P\left( {A|{E_1}} \right)P\left( {{E_1}} \right)}}{{P\left( {A|{E_1}} \right)P\left( {{E_1}} \right) + P\left( {A|{E_2}} \right)P\left( {{E_2}} \right)}}$
$= \cfrac{{0.001 \times 0.99}}{{0.001 \times 0.99 + 0.999 \times 0.005}}$
$= \cfrac{{990}}{{5985}} = \cfrac{{198}}{{1197}} = \cfrac{{22}}{{133}}$
NCERT & Exemplar solution for CBSE Class 12 Mathematics, Probability. Curated by Sachin Sharma. Free for all students.