Application of Bayes' Theorem: Find the probability that a person followed a course on meditation and yoga given that he suffers a heart attack.

Question

The Question:

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 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?

The official solution given is this:

Let $$A: \text{Person suffers a heart attack}$$ $$E_{1}: \text{Person follows a course on Meditation & Yoga}$$ $$E_{2}: \text{Person is treated with drug}$$

Given,

• $\Pr(A)= 0.4$
• $\Pr(A|E_{1}) = 0.4 * 0.7 = 0.28$
• $\Pr(A|E_{2}) = 0.4 * 0.75 = 0.3$

Also given that meditation & yoga and drug prescription has equal probabilities i.e., $\Pr(E_{1}) = \Pr(E_{2}) = 0.5$

We have to find the probability a person followed a course on meditation & yoga given that he suffered a heart attack i.e., $\Pr(E_{1}|A)$ Using Bayes' Theorem, $$\Pr(E_{1}|A) = \frac{\Pr(E_{1})*\Pr(A|E_{1})}{\Pr(E_{1})*\Pr(A|E_{1})+\Pr(E_{2})*\Pr(A|E_{2})}$$ Substituting values, we get $$\boxed{\Pr(E_{1}|A) = \frac{14}{29}}$$

When I first looked at the question, this is how I solved it:

From Bayes' Theorem, we know that $$\Pr(E_{1}|A) = \frac{\Pr(E_{1})*\Pr(A|E_{1})}{\Pr(A)}$$

Now I substituted the values to get $\Pr(E_{1}|A) = \frac{7}{20}$. Note that I used $\Pr(A) = 0.4$ (and here is where I think the mistake lies).

I then drew this diagram to visualise what I was doing:

So looking at the image, this where I think I went wrong:

When I take $\Pr(A) = 0.4$, I also include the people who did not suffer a heart attack because they took the medication (aka those who benefited from the course\drug prescription).

In the official solution, they ignored the unshaded part of the left $0.4 * 1$ rectangle (in my diagram) because they did not suffer a heart attack. Am I right? If no, where am I going wrong?

Answer 1

If in this context a patient has probability $0.5$ to undergo a course in Meditation/Yoga and has probability $0.5$ to be treated with a drug then:$$P(A)=P(A\mid E_1)P(E_1)+P(A\mid E_2)P(E_2)=0.28\times0.5+0.3\times0.5=0.29$$where $A$ denotes the event that the patient will get a heart attack.

Further we have the equality:$$P(E_1\mid A)P(A)=P(A\cap E_1)=P(A\mid E_1)P(E_1)$$

Substituting $P(A)$, $P(E_1)$ and $P(A\mid E_1)$ we find:$$P(E_1\mid A)\times0.29=0.28\times0.5$$and conclude that:$$P(E_1\mid A)=\frac{14}{29}$$

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The conditional probabilities $P(A\mid E_i)$ are deduced from the data.

Answer 2

Yes, you found your mistake correctly. To understand it a bit differently,

If $A$ is the event of a patient suffering a heart attack, then $P(A) \ne 0.4$. Instead, $P(A|E_0) = 0.4$ where $E_0$ is the event where a patient chooses none of the two given options but as everyone chooses one of the two options, $P(E_0) = 0$.

**Probability of a patient suffering a heart attack,

$P(A) = \sum \limits_{i=0}^{2}P(E_i) \times P(A|E_i) = 0.29$

Then it follows that

$P(E_1|A) = \displaystyle \frac{0.5 \times 0.28}{0.29} = \frac{14}{29}$