Probability
Properties of Probability
Grade 12
Question:
<p>Given <em>P</em>(<em>A</em> ∪ <em>B</em>) = <em>P</em>(<em>A</em> ∩ <em>B</em>), then which of the following is correct?</p><p>(1) <em>P</em>(<em>A</em>) + <em>P</em>(<em>B</em>) = 2<em>P</em>(<em>A</em> ∩ <em>B</em>)</p><p>(2) <em>P</em>(<em>A</em>) = <em>P</em>(<em>B</em>)</p><p>(3) <em>A</em> and <em>B</em> are equally likely</p><p>(4) All of the above</p>
<p>(1) only</p>
<p>(1) and (2) only</p>
<p>(1), (2) and (3)</p>
<p>All of the above</p>
Step-by-Step Solution
Key Concept: Use the fundamental identity P(A ∪ B) = P(A) + P(B) - P(A ∩ B) and set it equal to P(A ∩ B) to derive relationships between P(A) and P(B).
<p><strong>Step 1:</strong> Apply the formula for union of events:<br>P(A ∪ B) = P(A) + P(B) - P(A ∩ B)</p><p><strong>Step 2:</strong> Use the given condition P(A ∪ B) = P(A ∩ B):<br>P(A ∩ B) = P(A) + P(B) - P(A ∩ B)</p><p><strong>Step 3:</strong> Simplify:<br>2P(A ∩ B) = P(A) + P(B)</p><p><strong>Step 4:</strong> Verify statement (1): P(A) + P(B) = 2P(A ∩ B) ✓ TRUE</p><p><strong>Step 5:</strong> Check if (2) and (3) must be true:<br>Statement (2) P(A) = P(B) is NOT necessarily true. For example: P(A) = 0.3, P(B) = 0.5, P(A ∩ B) = 0.4 satisfies 0.3 + 0.5 = 2(0.4), but P(A) ≠ P(B)<br>Statement (3) is false for the same reason.</p><p><strong>Step 6:</strong> Only statement (1) is correct.</p><p>∴ Answer: D (Only statement 1 is correct)</p>
Correct Answer: D