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Probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsProbability3 Marks
Question
A laboratory blood test is 99% effective in detecting a certain disease when its infection is 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 disease). If 0.1% of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?
✓

Answer

Let E1 and E2 denote the events that a person has a disease and a person has no disease, respectively.
E1 and E2 are complimentary to each other.
$\therefore$ P(E1) + P(E2) = 1
⇒ P(E2) = 1 - P(E1) = 1 - 0.001 = 0999
let A denote the event that the blood test result is positive.
$\therefore$ P(E1) = 0.1% = 0.001
Now,
$\text{P}\Big(\frac{\text{A}}{\text{E}_1}\Big)=90\%=0.99$
$\text{P}\Big(\frac{\text{A}}{\text{E}_2}\big)=0.5\%=0.005$
Using Baye's theorem, we get
Required probability $\text{P}\Big(\frac{\text{E}_1}{\text{A}}\Big)=\frac{\text{P}(\text{E}_1)\text{P}\Big(\frac{\text{A}}{\text{E}_1}\Big)}{\text{P}(\text{E}_1)\text{P}\Big(\frac{\text{A}}{\text{E}_1}\Big)+\text{P}(\text{E}_2)\text{P}\Big(\frac{\text{A}}{\text{E}_2}\Big)}$
$=\frac{0.001\times0.99}{0.001\times0.99+0.999\times0.005}$
$=\frac{990}{5985}=\frac{22}{133}$

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