Probability
Classical Probability
Grade 12
Question:
<p>In a certain city two newspapers A and B are published. It is known that 25% of the city population reads A and 20% reads B while 8% reads both A and B. It is also known that 30% of those who read A but not B look into advertisements and 40% of those who read B but not A look advertisements while 50% of those who read both A and B look into advertisements. Which of the following is/are correct?</p>
<p>(a) Percentage of those who read A but not B is 17%</p>
<p>(b) Percentage of those who read A but not B is 19%</p>
<p>(c) Percentage of the population who reads an advertisement is more than 13%</p>
<p>(d) Percentage of the population who reads an advertisement is less than 14%</p>
Step-by-Step Solution
Key Concept: Use set theory to partition the population into mutually exclusive groups (A only, B only, both), then apply conditional probability separately to each group to find the total percentage looking at ads.
<p><strong>Step 1: Find the population segments using inclusion-exclusion principle</strong></p><p>Let total population = 100</p><p>• Read A = 25</p><p>• Read B = 20</p><p>• Read both A and B = 8</p><p>• Read A only (A ∩ B') = 25 - 8 = 17</p><p>• Read B only (B ∩ A') = 20 - 8 = 12</p><p>• Read neither = 100 - (17 + 12 + 8) = 63</p><p><strong>Step 2: Calculate advertisements viewed in each segment</strong></p><p>• Among A only (17 people): 30% look at ads = 0.30 × 17 = 5.1</p><p>• Among B only (12 people): 40% look at ads = 0.40 × 12 = 4.8</p><p>• Among both (8 people): 50% look at ads = 0.50 × 8 = 4</p><p><strong>Step 3: Calculate total and conditional probabilities</strong></p><p>• Total looking at ads = 5.1 + 4.8 + 4 = 13.9 out of 100</p><p>• % of population viewing ads = 13.9%</p><p>• % of A readers viewing ads = (5.1 + 4)/(25) = 9.1/25 = 36.4%</p><p>• % of B readers viewing ads = (4.8 + 4)/(20) = 8.8/20 = 44%</p><p>• % among those reading exactly one = (5.1 + 4.8)/(29) = 9.9/29 ≈ 34.1%</p><p><strong>Verify which statements match these calculations (A, C, D are typically the correct options based on these probabilities)</strong></p>
Correct Answer: A,C,D