NPTEL Assignment Question

Question

A survey of people in given region showed that 25% drank regularly. The probability of death due to liver disease, given that a person drank regularly, was 6 times the probability of death due to liver disease, given that a person did not drink regularly. The probability of death due to liver disease in the region is 0.005. If a person dies due to liver disease what is the probability that he/she drank regularly?

Answer 1

\( P(\text{Drank Regularly}) = 25\% = 0.25 \)
\( P(\text{Died} \mid \text{Drank Regularly}) = 6 \times P(\text{Died} \mid \text{Did Not Drink Regularly}) \)
\( P(\text{Died due to Liver Disease}) = 0.005 \)
\[P(\text{Died} \mid \text{Did Not Drink Regularly}) = x\]
\[P(\text{Died} \mid \text{Drank Regularly}) = 6x\]

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Law of Total Probability,
\[0.005 = (6x) \cdot 0.25 + x \cdot 0.75\]
\[0.005 = 1.5x + 0.75x\]
\[0.005 = 2.25x\]
\[x = \frac{0.005}{2.25} = \frac{1}{450}\]
\[P(\text{Died} \mid \text{Drank Regularly}) = 6x = \frac{6}{450} = \frac{1}{75}\]
\[P(\text{Died} \mid \text{Did Not Drink Regularly}) = x = \frac{1}{450}\]
Bayes' Theorem,
\[P(\text{Drank Regularly} \mid \text{Died}) = \frac{P(\text{Died} \mid \text{Drank Regularly}) \cdot P(\text{Drank Regularly})}{P(\text{Died})}\]
\[P(\text{Drank Regularly} \mid \text{Died}) = \frac{\frac{1}{75} \cdot 0.25}{0.005} = \frac{\frac{0.25}{75}}{0.005} = \frac{0.003333}{0.005} = 0.6667\]
\[\boxed{\dfrac{2}{3}}\]