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 drugs 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 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. So, P (A) = 0.40, P (E1​) = P (E2​) = $\dfrac{1}{2}$. Find P(A ∣ E1​) and P(A ∣ E2​). Find P(E1​ | A) using the formula:$P({{E}_{1}}|A)=\dfrac{P({{E}_{1}})P(A|{{E}_{1}})}{P({{E}_{1}})P(A|{{E}_{1}})+P({{E}_{2}})P(A|{{E}_{2}})}$. Substitute the above probabilities and solve to get the final answer.

Complete step-by-step answer :
In this question, we are given 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 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.
We need to 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.
So, P (A) = 0.40
P (E1​) = P (E2​) = $\dfrac{1}{2}$
Now, it is given, that if the patient does meditation and yoga, it reduces the risk by 30%. So, the risk becomes 70% . It is also given that if the patient takes prescription drugs, it reduces the risk by 25%. So, the risk in this case becomes 75%.

Now, we will use the concept of conditional probability. The conditional probability of an event B is the probability that the event will occur given the knowledge that an event A has already occurred. This probability is written P(B|A), notation for the probability of B given A.
Now, P(A|${{E}_{1}}$ ) indicates the probability of having heart attack despite doing yoga and meditation, which can be given as P(A) $\times$ Risk.
$\Rightarrow$ P(A|${{E}_{1}}$ ) = 0.40 $\times$ 0.70 = 0.28
Similarly, P(A|${{E}_{2}}$ ) indicates the probability of having heart attack despite doing yoga and meditation, which can be given as P(A) $\times$ Risk.
$\Rightarrow$ P(A|${{E}_{2}}$ ) = 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( ​∣ A).
Using the law of total probability, we have the following:
$P({{E}_{1}}|A)=\dfrac{P({{E}_{1}})P(A|{{E}_{1}})}{P({{E}_{1}})P(A|{{E}_{1}})+P({{E}_{2}})P(A|{{E}_{2}})}$
$P({{E}_{1}}|A)=\dfrac{\dfrac{1}{2}\times 0.28}{\dfrac{1}{2}\times 0.28+\dfrac{1}{2}\times 0.30}=\dfrac{14}{29}=0.48$
Hence, the probability that the patient suffering a heart attack followed a course of meditation and yoga is 0.48.

Note :In this question, it is very important to know about conditional probability. Conditional probability is a measure of the probability of an event occurring given that another event has occurred. The conditional probability of A given B is $P(A|B)=\dfrac{P(A\cap B)}{P(B)}$.