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 for0.5^{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 ?

Answers (1)

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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