Question

Assume that the chances of a patient having a heart attack is 40%. Assuming that a meditation and yoga course reduces the risk of heart attack by 30% and prescription of certain drugs 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 meditation and yoga. Interpret the result and state which of the above stated methods is more beneficial for the patient.

Answer 1

We start solving the above problem by considering the event that the person has a heart attack as $A$, the event that the person is treated by meditation and yoga as ${{E}_{1}}$ and the event that the person is treated by drugs as ${{E}_{2}}$. Then we calculate the probability of occurring of the event ${{E}_{1}}$ and the probability of occurring of the event ${{E}_{2}}$. Then we find the probability of having a heart attack if the person is treated with meditation and yoga and the probability of having a heart attack if the person is treated with drugs. Finally, we find the probability that the person suffering from heart attack followed a course of meditation and yoga using the formula $P\left( {{{E}_{1}}}/{A}\; \right)=\dfrac{P\left( {{E}_{1}} \right).P\left( {A}/{{{E}_{1}}}\; \right)}{P\left( {{E}_{1}} \right).P\left( {A}/{{{E}_{1}}}\; \right)+P\left( {{E}_{2}} \right).P\left( {A}/{{{E}_{2}}}\; \right)}$. Then we interpret the result and state which of the above stated methods is more beneficial for the patient.

Complete step by step answer:
Let us consider the event that the person has heart attack as event $A$,
the event that the person is treated by meditation and yoga as event ${{E}_{1}}$
and the event that the person is treated by drugs as ${{E}_{2}}$.
We were given that the normal risk of heart attack is $40\%$.
So, we get,
$\begin{aligned} & P\left( A \right)=40\% \\\ & \Rightarrow P\left( A \right)=\dfrac{40}{100} \\\ & \Rightarrow P\left( A \right)=0.4 \\\ \end{aligned}$
We were also given that the meditation and yoga and drugs have equal probabilities.
So, as the sum of the probabilities is always 1, we get,
$P\left( {{E}_{1}} \right)+P\left( {{E}_{2}} \right)=1$
As we were given $P\left( {{E}_{1}} \right)=P\left( {{E}_{2}} \right)$, we get,
$\begin{aligned} & P\left( {{E}_{1}} \right)+P\left( {{E}_{1}} \right)=1 \\\ & \Rightarrow 2P\left( {{E}_{1}} \right)=1 \\\ & \Rightarrow P\left( {{E}_{1}} \right)=\dfrac{1}{2} \\\ \end{aligned}$
So, $P\left( {{E}_{1}} \right)=P\left( {{E}_{2}} \right)=\dfrac{1}{2}$.
$\therefore P\left( {{E}_{1}} \right)=0.5$ and $P\left( {{E}_{2}} \right)=0.5$
Now, let us consider the probability of having heart attack if the person is treated with meditation and yoga, that is, $P\left( {A}/{{{E}_{1}}}\; \right)$
We were given that meditation and yoga reduce the risk of heart attack by $30\%$. It means $70\%$ face the risk.
So, we get,
$\begin{aligned} & \Rightarrow P\left( {A}/{{{E}_{1}}}\; \right)=0.40\times 70\% \\\ & \Rightarrow P\left( {A}/{{{E}_{1}}}\; \right)=0.40\times \dfrac{70}{100} \\\ & \Rightarrow P\left( {A}/{{{E}_{1}}}\; \right)=0.40\times 0.70 \\\ & \Rightarrow P\left( {A}/{{{E}_{1}}}\; \right)=0.28 \\\ \end{aligned}$
Now, let us consider the probability of having heart attack if the person is treated with meditation and yoga, that is, $P\left( {A}/{{{E}_{2}}}\; \right)$
As we were given that the drugs reduce the risk of heart attack by $25$, it means $75\%$ face the risk. So, we get,
$\begin{aligned} & P\left( {A}/{{{E}_{2}}}\; \right)=0.40\times 75\% \\\ & \Rightarrow P\left( {A}/{{{E}_{2}}}\; \right)=0.40\times \dfrac{75}{100} \\\ & \Rightarrow P\left( {A}/{{{E}_{2}}}\; \right)=0.40\times 0.75 \\\ & \Rightarrow P\left( {A}/{{{E}_{2}}}\; \right)=0.30 \\\ \end{aligned}$
Now, we consider the formula of Bayes theorem, that is,
$P\left( {{{E}_{1}}}/{A}\; \right)=\dfrac{P\left( {{E}_{1}} \right).P\left( {A}/{{{E}_{1}}}\; \right)}{P\left( {{E}_{1}} \right).P\left( {A}/{{{E}_{1}}}\; \right)+P\left( {{E}_{2}} \right).P\left( {A}/{{{E}_{2}}}\; \right)}$.

By using the above formula, we get,

& P\left( {{{E}_{1}}}/{A}\; \right)=\dfrac{0.5\times 0.28}{\left( 0.5\times 0.28 \right)+\left( 0.5\times 0.30 \right)} \\\ & \Rightarrow P\left( {{{E}_{1}}}/{A}\; \right)=\dfrac{0.5\times 0.28}{0.5\left( 0.28+0.30 \right)} \\\ & \Rightarrow P\left( {{{E}_{1}}}/{A}\; \right)=\dfrac{0.28}{0.28+0.30} \\\ & \Rightarrow P\left( {{{E}_{1}}}/{A}\; \right)=\dfrac{0.28}{0.58} \\\ & \Rightarrow P\left( {{{E}_{1}}}/{A}\; \right)=\dfrac{14}{29} \\\ \end{aligned}$$ **Hence, the probability that the patient followed a course of meditation and yoga had heart attack is $\dfrac{14}{29}$. Now let us look at the probability of having heart attack if the person is treated with meditation and yoga, as we see $P\left( {A}/{{{E}_{1}}}\; \right)$ is less than the probability of having heart attack if the person is treated with meditation and yoga, that is, $P\left( {A}/{{{E}_{2}}}\; \right)$. Therefore, we can conclude that the course of following meditation and yoga is beneficial when compared to the course of drugs.** **Note:** One may make a mistake while calculating the probability of having heart attack if the person is treated with meditation and yoga, that is, $P\left( {A}/{{{E}_{1}}}\; \right)$ by multiplying it with 30% instead of 70%, and while calculating the probability of having heart attack if the person is treated with drugs, that is, $P\left( {A}/{{{E}_{2}}}\; \right)$ by multiplying it with 25% instead of 75%. Here we are calculating the probability for people attacked by heart attack not the people that escaped it. So, one needs to be careful while calculating it.