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Assume that the chances of a patient having a heart attack is $40 \%$. It is also assumed that a meditation and yoga course reduce the risk of heart attack by $30 \%$ and prescription of certain drug reduces it 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?
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A patient has options to have the treatment of Yoga and Meditation and that of prescription of drugs.
Let these events be denoted by $\mathrm{E}_1$ and $\mathrm{E}_2$ i.e.,
$\mathrm{E}_1$ : Treatment of Yoga and Mediation.
$\mathrm{E}_2$ : Treatment of prescription of certain drugs;
$$
\mathrm{P}\left(\mathrm{E}_1\right)=\frac{1}{2}, \mathrm{P}\left(\mathrm{E}_2\right)=\frac{1}{2}
$$
Let A denotes that a person has heart attack,
$$
\mathrm{P}(\mathrm{A})=40 \%=0-40
$$
Yoga and Meditation reduces the heart risk by $30 \%$
$\Rightarrow$ inspite of getting Yoga and Meditiation treatment heart risk is $70 \%$ of the $0.40$
$$
\Rightarrow \quad \mathrm{P}\left(\mathrm{A} / \mathrm{E}_1\right)=0.40 \times 0.70=0.28
$$
Drug prescription reduces the heart risk by $25 \%$ Even after adopting the drug prescription heart rist is $75 \%$ of the $0.40$ $\therefore \quad \mathrm{P}\left(\mathrm{A} / \mathrm{E}_2\right)=0.40 \times 0.75=0.30 ;$ Now we have
$$
\begin{aligned}
&\mathrm{P}\left(\mathrm{E}_1\right)=\frac{1}{2}, \mathrm{P}\left(\mathrm{E}_2\right)=\frac{1}{2} \\
&\mathrm{P}\left(\mathrm{A} / \mathrm{E}_1\right)=0.28, \mathrm{P}\left(\mathrm{A} / \mathrm{E}_2\right)=0 \cdot 30 \\
&\mathrm{P}\left(\mathrm{E}_1 / \mathrm{A}\right)=\frac{\mathrm{P}\left(\mathrm{E}_1\right) \mathrm{P}\left(\mathrm{A} / \mathrm{E}_1\right)}{\mathrm{P}\left(\mathrm{E}_1\right) \mathrm{P}\left(\mathrm{A} / \mathrm{E}_1\right)+\mathrm{P}\left(\mathrm{E}_2\right) \mathrm{P}\left(\mathrm{A} / \mathrm{E}_2\right)}
\end{aligned}
$$
$$
=\frac{\frac{1}{2} \times 0.28}{\frac{1}{2} \times 0.28+\frac{1}{2} \times 0.30}=\frac{0.28}{0.58}=\frac{28}{58}=\frac{14}{29}
$$
Let these events be denoted by $\mathrm{E}_1$ and $\mathrm{E}_2$ i.e.,
$\mathrm{E}_1$ : Treatment of Yoga and Mediation.
$\mathrm{E}_2$ : Treatment of prescription of certain drugs;
$$
\mathrm{P}\left(\mathrm{E}_1\right)=\frac{1}{2}, \mathrm{P}\left(\mathrm{E}_2\right)=\frac{1}{2}
$$
Let A denotes that a person has heart attack,
$$
\mathrm{P}(\mathrm{A})=40 \%=0-40
$$
Yoga and Meditation reduces the heart risk by $30 \%$
$\Rightarrow$ inspite of getting Yoga and Meditiation treatment heart risk is $70 \%$ of the $0.40$
$$
\Rightarrow \quad \mathrm{P}\left(\mathrm{A} / \mathrm{E}_1\right)=0.40 \times 0.70=0.28
$$
Drug prescription reduces the heart risk by $25 \%$ Even after adopting the drug prescription heart rist is $75 \%$ of the $0.40$ $\therefore \quad \mathrm{P}\left(\mathrm{A} / \mathrm{E}_2\right)=0.40 \times 0.75=0.30 ;$ Now we have
$$
\begin{aligned}
&\mathrm{P}\left(\mathrm{E}_1\right)=\frac{1}{2}, \mathrm{P}\left(\mathrm{E}_2\right)=\frac{1}{2} \\
&\mathrm{P}\left(\mathrm{A} / \mathrm{E}_1\right)=0.28, \mathrm{P}\left(\mathrm{A} / \mathrm{E}_2\right)=0 \cdot 30 \\
&\mathrm{P}\left(\mathrm{E}_1 / \mathrm{A}\right)=\frac{\mathrm{P}\left(\mathrm{E}_1\right) \mathrm{P}\left(\mathrm{A} / \mathrm{E}_1\right)}{\mathrm{P}\left(\mathrm{E}_1\right) \mathrm{P}\left(\mathrm{A} / \mathrm{E}_1\right)+\mathrm{P}\left(\mathrm{E}_2\right) \mathrm{P}\left(\mathrm{A} / \mathrm{E}_2\right)}
\end{aligned}
$$
$$
=\frac{\frac{1}{2} \times 0.28}{\frac{1}{2} \times 0.28+\frac{1}{2} \times 0.30}=\frac{0.28}{0.58}=\frac{28}{58}=\frac{14}{29}
$$
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