Assume that the chances of the patient having a heart attack are 40%. It is also assumed that a meditation and yoga course reduce 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 the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?
Let A, E1, and E2 respectively denote the events that a person has a heart attack, the selected person followed the course of yoga and meditation, and the person adopted the drug prescription.
$\therefore \mathrm{P}(\mathrm{A})=0.40$
$P\left(E_{1}\right)=P\left(E_{2}\right)=\frac{1}{2}$
$\mathrm{P}\left(\mathrm{A} \mid \mathrm{E}_{1}\right)=0.40 \times 0.70=0.28$
$\mathrm{P}\left(\mathrm{A} \mid \mathrm{E}_{2}\right)=0.40 \times 0.75=0.30$
Probability that the patient suffering a heart attack followed a course of meditation and yoga is given by P (E1|A).
$P\left(E_{1} \mid A\right)=\frac{P\left(E_{1}\right) P\left(A \mid E_{1}\right)}{P\left(E_{1}\right) P\left(A \mid E_{1}\right)+P\left(E_{2}\right) P\left(A \mid E_{2}\right)}$
$=\frac{\frac{1}{2} \times 0.28}{\frac{1}{2} \times 0.28+\frac{1}{2} \times 0.30}$
$=\frac{14}{29}$