Probability
Bayes' Theorem
Grade 12
Question:
<p>The probability that a particular day in the month of July is a rainy day is 3/4. Two person whose credibility are 4/5 and 2/3, respectively, claim that 15th July was a rainy day. Find the probability that it was really a rainy day.</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem with independent witness testimonies: P(Rain|both claim rain) = P(both claim|Rain)×P(Rain) / P(both claim). The credibility of each witness means P(claim rain|actually rain) = credibility, and P(claim rain|not rain) = 1 - credibility.
<p><strong>Step 1:</strong> Define events. Let R = it rained, C = both claim it rained.</p><p><strong>Step 2:</strong> Prior probability: P(R) = 3/4, P(R') = 1/4</p><p><strong>Step 3:</strong> Likelihood if it rained: P(C|R) = (4/5) × (2/3) = 8/15 (both correctly claim rain)</p><p><strong>Step 4:</strong> Likelihood if it didn't rain: P(C|R') = (1 - 4/5) × (1 - 2/3) = (1/5) × (1/3) = 1/15 (both falsely claim rain)</p><p><strong>Step 5:</strong> Total probability of both claiming rain: P(C) = P(C|R)×P(R) + P(C|R')×P(R') = (8/15)×(3/4) + (1/15)×(1/4) = 24/60 + 1/60 = 25/60 = 5/12</p><p><strong>Step 6:</strong> Apply Bayes' theorem: P(R|C) = P(C|R)×P(R) / P(C) = (8/15 × 3/4) / (5/12) = (24/60) / (5/12) = (2/5) × (12/5) = 24/25</p><p>∴ Answer: <strong>24/25</strong></p>
Correct Answer: 24/25