# Q. 5.    A laboratory blood test is $99^{o}/_{o}$ effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for$0.5^{o}/_{o}$ 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   $0.1^{o}/_{o}$  of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive ?

S seema garhwal

A  : Person selected is  having the disease

B : Person selected is not having the disease.

C :Blood result is positive.

$P(A)= 0.1 \%=\frac{1}{1000}=0.001$

$P(B)= 1 -P(A)=1-0.001=0.999$

$P(C|A)=99\%=0.99$

$P(C|B)=0.5\%=0.005$

By Bayes theorem :

$P(A|C)=\frac{P(A).P(C|A)}{P(A).P(C|A)+P(B).P(C|B)}$

$=\frac{0.001\times 0.99}{0.001\times 0.99+0.999\times 0.005}$

$=\frac{0.00099}{0.00099+0.004995}$

$=\frac{0.00099}{0.005985}$ $=\frac{990}{5985}$

$=\frac{22}{133}$

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