Q.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?

Concept: Conditional Probability (Bayes' Theorem)

Let $H$ be the event of a heart attack, $M$ the event of choosing meditation & yoga, and $D$ the event of choosing the drug.

Step 1: Prior probabilities

$P(M) = P(D) = 0.5$ (equal choice).

Step 2: Conditional risks

Original risk of heart attack = $0.4$.

• Meditation reduces risk by $30\%$, so $P(H \mid M) = 0.4 \times (1 - 0.3) = 0.28$.
• Drug reduces risk by $25\%$, so $P(H \mid D) = 0.4 \times (1 - 0.25) = 0.30$.

Step 3: Apply Bayes' Theorem

We need $P(M \mid H)$: …

Let $A_1$ = "chose meditation & yoga", $A_2$ = "chose the drug", and $H$ = "suffers a heart attack". The patient picks either option with equal probability:

1. Conditional risks. The baseline chance is $0.40$. A reduction of $30\%$ leaves $70\%$ of the risk, and a reduction of $25\%$ leaves $75\%$:

$P(H\mid A_2)=0.40\times(1-0.25)=0.40\times0.75=0.30.$ …

Method: Bayes' Theorem after adjusting a baseline risk

Use this when each option (treatment) changes a baseline probability by a stated percentage, and you must find, given the outcome occurred, which option was chosen.

Steps

Step 1: Turn each "reduces risk by r%" into an actual conditional probability.

A reduction of $r\%$ is relative to the baseline $b$: it leaves $(1-r)$ of it, so $P(E \mid \text{option}) = b \times (1 - r)$. It is not $b - r$. Do this for every option before touching Bayes' theorem.

Step 2: Record the priors of the options.

$P(H_i)$ — equal, if the patient picks any option with equal probability.

Step 3: Total probability of the outcome. …

Common Mistakes

Mistake 1: Reading "reduces risk by 30%" as an absolute subtraction.

Why it's wrong: it is a relative reduction of the $0.4$ baseline, so the risk becomes $0.4\times0.7=0.28$, not $0.4-0.3=0.1$. Correct approach: multiply the baseline by $(1-\text{reduction})$ for each option.

Mistake 2: Applying the reduction to only one option.

Why it's wrong: both the yoga and the drug conditional risks must be recomputed. Correct approach: $P(H\mid \text{yoga})=0.28$ and $P(H\mid \text{drug})=0.4\times0.75=0.30$. …