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Question
Assume that the chances of a patient having a heart attack is 40%. It is also assumed that meditation and yoga course reduces the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options and patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?
Solution
Let A, E1 and E2 denote the events that the selected person had a heart attack, did yoga and meditation, and followed the drug prescriptions, respectively.
\[\therefore P\left( E_1 \right) = \frac{1}{2} \]
\[ P\left( E_2 \right) = \frac{1}{2}\]
\[\text{ Now, } \]
\[P\left( A/ E_1 \right) = 0.40 \times 0.70 = 0 .28\]
\[P\left( A/ E_2 \right) = 0.40 \times 0.75 = 0.30\]
\[\text{ Using Bayes' theorem, we get} \]
\[\text{ Required probability } = P\left( E_1 /A \right) = \frac{P\left( E_1 \right)P\left( A/ E_1 \right)}{P\left( E_1 \right)P\left( A/ E_1 \right) + P\left( E_2 \right)P\left(A/ E_2 \right)}\]
\[ = \frac{\frac{1}{2} \times 0.28}{\frac{1}{2} \times 0.28 + \frac{1}{2} \times 0.30}\]
\[ = \frac{14}{29}\]
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