Question

Assume that the chances of a patient having a heart attack is$40\%$. It is also assumed that a 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 the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?

Answer 1

Let $E_1$: the event that the patient follows meditation and yoga and

$E_2$: the event that the patient uses drug.

$\therefore E_{1}and{E}_2$ are mutually exclusive events and $P\left(E_1\right)=P\left(E_2\right)=\frac{1}{2}$

Let E: the event that the selected patient suffer a heart attack

$\therefore P\left(\frac{E}{E_1}\right)=\frac{40}{100}(1-\frac{30}{100})=\frac{28}{100}P\left(\frac{E}{E_2}\right)=\frac{40}{100}(1-\frac{25}{100})=\frac{30}{100}$

$\therefore$ P (patient who suffers heart attack follows meditation and yoga)

$P\left(\frac{E}{E_1}\right)=\frac{P\left(\frac{E}{E_1}\right)P\left(E_1\right)}{P\left(\frac{E}{E_1}\right)P\left(E_1\right)+P\left(\frac{E}{E_2}\right)P\left(E_2\right)}=\frac{\frac{28}{100}\times \frac{1}{2}}{\frac{28}{100}\times \frac{1}{2}+\frac{30}{100}\times \frac{1}{2}}$

$=\frac{\frac{14}{100}}{\frac{14}{100}+\frac{15}{100}}=\frac{\frac{14}{100}}{\frac{14}{100}+\frac{15}{100}}=\frac{14}{100}\times \frac{100}{29}=\frac{14}{29}$