Probability
Bayes Theorem
Grade 12
Question:
<p>A purse contains 2 six-sided dice. One is a normal fair die, while the other has two 1's, two 3's, and two 5's. A die is picked up and rolled. Because of some secret magnetic attraction of the unfair die, there is 75% chance of picking the unfair die and a 25% chance of picking a fair die. The die is rolled and shows up the face 3. The probability that a fair die was picked up is</p>
<p>(1) 1/7</p>
<p>(2) 1/4</p>
<p>(3) 1/6</p>
<p>(4) 1/24</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem to find P(Fair | Face 3) by calculating the probability of getting 3 from each die and applying the conditional probability formula with the given prior probabilities.
<p><strong>Step 1: Identify the probabilities</strong></p><p>Prior probabilities: P(Fair) = 0.25, P(Unfair) = 0.75</p><p>The unfair die has two 1's, two 3's, and two 5's (no even faces).</p><p><strong>Step 2: Calculate likelihoods</strong></p><p>P(Face 3 | Fair die) = 1/6</p><p>P(Face 3 | Unfair die) = 2/6 = 1/3</p><p><strong>Step 3: Apply Bayes' Theorem</strong></p><p>P(Fair | Face 3) = [P(Face 3 | Fair) × P(Fair)] / P(Face 3)</p><p>where P(Face 3) = P(Face 3 | Fair) × P(Fair) + P(Face 3 | Unfair) × P(Unfair)</p><p>P(Face 3) = (1/6)(0.25) + (1/3)(0.75) = 1/24 + 6/24 = 7/24</p><p><strong>Step 4: Calculate posterior probability</strong></p><p>P(Fair | Face 3) = [(1/6)(0.25)] / (7/24) = (1/24) / (7/24) = 1/7</p><p>∴ Answer: 1/7</p>
Correct Answer: 1