Probability
Bayes' Theorem
Grade 12
Question:
<p>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?</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem to find P(Meditation|Heart Attack) by first calculating the probability of heart attack under each option, then applying the conditional probability formula.
<p><strong>Step 1: Identify probabilities for each option</strong></p><p>Initial heart attack probability: P(H) = 0.40</p><p>• If Meditation & Yoga (M): Risk reduces by 30% → P(H|M) = 0.40 × (1 - 0.30) = 0.40 × 0.70 = 0.28</p><p>• If Drug (D): Risk reduces by 25% → P(H|D) = 0.40 × (1 - 0.25) = 0.40 × 0.75 = 0.30</p><p>• Each option chosen with probability: P(M) = P(D) = 0.5</p><p><strong>Step 2: Apply the law of total probability</strong></p><p>P(H) = P(H|M)·P(M) + P(H|D)·P(D)</p><p>P(H) = 0.28 × 0.5 + 0.30 × 0.5 = 0.14 + 0.15 = 0.29</p><p><strong>Step 3: Apply Bayes' theorem</strong></p><p>P(M|H) = P(H|M)·P(M) / P(H)</p><p>P(M|H) = (0.28 × 0.5) / 0.29 = 0.14 / 0.29 = 14/29</p><p><strong>∴ Answer: 14/29</strong></p>
Correct Answer: 14/29