Probability
Addition theorem and conditional probability
Grade 12
Question:
<p>In a certain city, only 2 newspapers <em>A</em> and <em>B</em> are published. It is known that 25% of the city population read <em>A</em> and 20% read <em>B</em> while 8% reads both <em>A</em> and <em>B</em>. It is also known that 30% of those who read <em>A</em> but not <em>B</em> look into advertisement and 40% of those who read <em>B</em> but not <em>A</em> look into advertisements while 50% of those who read both <em>A</em> and <em>B</em> look into advertisements. What is the percentage of the population who read an advertisement?</p>
<p>13.9%</p>
<p>12.8%</p>
<p>15.0%</p>
<p>10.5%</p>
Step-by-Step Solution
Key Concept: Break the population into mutually exclusive groups (A only, B only, both A and B) using set theory, then apply the law of total probability by weighting each group's advertisement-reading percentage by its proportion of total population.
<p><strong>Step 1: Partition the population into mutually exclusive groups</strong></p><p>• Read A only: 25% − 8% = 17%</p><p>• Read B only: 20% − 8% = 12%</p><p>• Read both A and B: 8%</p><p>• Read neither: 100% − (17% + 12% + 8%) = 63%</p><p><strong>Step 2: Find the percentage who read advertisements in each group</strong></p><p>• From A only: 17% × 30% = 17% × 0.30 = 5.1%</p><p>• From B only: 12% × 40% = 12% × 0.40 = 4.8%</p><p>• From both: 8% × 50% = 8% × 0.50 = 4%</p><p><strong>Step 3: Apply the law of total probability</strong></p><p>Total percentage reading advertisements = 5.1% + 4.8% + 4% = <strong>13.9%</strong></p><p>∴ Answer: A</p>
Correct Answer: A