Probability
Probability Rules and Bounds
Grade 12
Question:
<p><strong>Question 1:</strong> If \(P(A) = 0.8\), \(P(B) = 0.5\), then \(P(A \cap B)\) lies in the interval</p>
<p>(a) \([0.2, 0.5]\)</p>
<p>(b) \([0.2, 0.3]\)</p>
<p>(c) \([0.3, 0.5]\)</p>
<p>(d) \([0.1, 0.5]\)</p>
Step-by-Step Solution
Key Concept: The intersection of two events is bounded by the constraint that P(A∪B) ≤ 1 from below and min{P(A), P(B)} from above.
<p><strong>Solution:</strong></p><p>We use the formula: $P(A \cap B) \leq \min\{P(A), P(B)\}$</p><p>Also, $P(A \cap B) \geq P(A) + P(B) - 1$ (from $P(A \cup B) \leq 1$)</p><p>Lower bound: $P(A \cap B) \geq 0.8 + 0.5 - 1 = 0.3$</p><p>Upper bound: $P(A \cap B) \leq \min\{0.8, 0.5\} = 0.5$</p><p>Therefore, $P(A \cap B) \in [0.3, 0.5]$</p><p>∴ Answer is (c)</p>
Correct Answer: a