Question

Assume that the chances of a patient having a heart attack is 40%. Assuming that a mediation and yoga course reduces the risk of heart attack by 30% and prescription of certain drug reduces its chance by 25%. At a time a patient can choose any one of the two options with equal possibilities. 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 mediation and yoga. Interpret the result and state which of the above stated methods is more beneficial for the patient.

Answer 1

Let E be the event that a patient had a heart attack.
Let $E_1$ be the event that a patient used Drug A, therefore, $P\left(E_1\right)=\frac{1}{2}$

Let $E_2$ be the event that a patient used Drug B, therefore, $P\left(E_2\right)=\frac{1}{2}$

Required probability: $P\left(\frac{E_1}{E}\right)$
$P\left(\frac{E}{E_1}\right)=\frac{40}{100}(1-\frac{30}{100})=\frac{28}{100}$

$P\left(\frac{E}{E_2}\right)=\frac{40}{100}(1-\frac{25}{100})=\frac{30}{100}$

Also $P\left(E_1\right)=P\left(E_2\right)=\frac{1}{2}$
$P($Probability that the person who had heart attach followed meditation and yoga$)=$ $P\left(\frac{E_1}{E}\right)$

$P\left(\frac{E_2}{E}\right)=\frac{P\left(\frac{E}{E_2}\right).P\left(E_2\right)}{\sum _{i=1}^4\left(P\left(\frac{E}{E_i}\right)\right).P\left(E_i\right)}$

$=\frac{\frac{28}{100}\times \frac{1}{2}}{\frac{28}{100}\times \frac{1}{2}+\frac{30}{100}\times \frac{1}{2}}$

$=\frac{\frac{28}{200}}{\frac{28}{200}+\frac{30}{200}}=\frac{28}{28+30}=\frac{14}{29}$