Q78 / 160 · TS EAMCET 2020 (14 Sep Shift 2) · Mathematics

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
    0.9545
  • B
    0.2194
  • Correct answer
    C
    0.0455
  • D
    0.9499

Solution

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 $\begin{aligned} & P\left(E_1 / A\right)=\frac{P\left(E_1\right) \times P\left(A / E_1\right)}{P\left(E_1\right) \times P\left(A / E_1\right)+P\left(E_2\right) \times P\left(A / E_2\right)} \\ & =\frac{0.005 \times 0.95}{(0.005 \times 0.95)+0.995 \times 0.10} \\ & \quad=\frac{0.00475}{0.00475+0.0995}=\frac{0.00475}{0.10425}=\frac{475}{10425}=0.0455 \end{aligned}$

Hence, the correct answer is option: 'C'.

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