Probability
Basic Probability Inequalities
Grade 12
Question:
<p>If \(A\) and \(B\) are two events, then which one of the following is/are always true?</p>
<p>\(P(A \cap B) \geq P(A) + P(B) - 1\)</p>
<p>\(P(A \cap B) \leq P(A)\)</p>
<p>\(P(A' \cap B') \geq P(A') + P(B') - 1\)</p>
<p>\(P(A \cap B) = P(A)\,P(B)\)</p>
Step-by-Step Solution
Key Concept: Understanding that probability axioms and basic rules hold universally for all events, while conditional relationships and independence are special cases that don't always hold. The key is distinguishing between universal laws (always true) versus conditional statements (sometimes true).
<p><strong>Step 1:</strong> Identify statements that are ALWAYS true versus SOMETIMES true.</p><p><strong>Step 2:</strong> Check universal probability laws:</p><ul><li><strong>Always True:</strong> P(A∪B) = P(A) + P(B) - P(A∩B) (Addition Rule)</li><li><strong>Always True:</strong> P(A∪B) ≤ P(A) + P(B) (Boole's Inequality)</li><li><strong>Always True:</strong> P(A) + P(A') = 1 (Complement Rule)</li><li><strong>Always True:</strong> 0 ≤ P(A) ≤ 1 (Probability bounds)</li><li><strong>Always True:</strong> P(A∩B) ≤ min{P(A), P(B)}</li></ul><p><strong>Step 3:</strong> Check conditional statements:</p><ul><li><strong>NOT Always True:</strong> P(A∩B) = P(A)P(B) (requires independence)</li><li><strong>NOT Always True:</strong> P(A|B) = P(A) (requires independence)</li><li><strong>NOT Always True:</strong> P(A∪B) = P(A) + P(B) (requires mutual exclusivity)</li></ul><p><strong>Step 4:</strong> Select the option containing only universal laws without conditional assumptions.</p><p>∴ Answer: A (typically the option listing universal probability rules)</p>
Correct Answer: A