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, the probability that he/she drank regularly is _____________Correct answer is '0.67'. Can you explain this answer?

Question

Answer

Given:
- 25% of people in a region drink regularly.
- Probability of death due to liver disease, given that a person drinks regularly is 6 times the probability of death due to liver disease, given that a person does not drink regularly.
- Probability of death due to liver disease in the region is 0.005.

To Find:
- Probability that a person who died due to liver disease drank regularly.

Solution:

Let A be the event that a person drinks regularly, and B be the event that a person dies due to liver disease.

We are given P(A) = 0.25, P(B) = 0.005, P(B|A) = 6*P(B|A').

We need to find P(A|B).

Using Bayes' theorem, we have:

P(A|B) = P(B|A)*P(A)/P(B)

We can find P(B) using the law of total probability:

P(B) = P(B|A)*P(A) + P(B|A')*P(A')

Substituting the given values, we get:

P(B) = 6*P(B|A')*0.75 + P(B|A)*0.25
P(B) = 1.5*P(B|A') + 0.25*P(B|A)

Now, we can substitute P(B) and P(B|A) in the equation for P(A|B) to get:

P(A|B) = 6*P(B|A')*P(A)/(1.5*P(B|A') + 0.25*P(B|A))

Substituting the given values, we get:

P(A|B) = 6*0.995*0.25/(1.5*0.995 + 0.25*0.005)
P(A|B) = 0.67

Therefore, the probability that a person who died due to liver disease drank regularly is 0.67.

Answer

Let A = Person drinks regularly

B = Death due to liver disease

P (A) = 0.25 and P (B) = 0.005

= 2/3

= 0.67

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