<p>(1) \(P_{100} > 2/3\)</p>
<p>(2) \(P_{101} < 2/3\)</p>
<p>(3) \(P_{100}, P_{101} > 2/3\)</p>
<p>(4) none of these</p>
Step-by-Step Solution
Key Concept: A statement is 'not true' if it's either false or doesn't hold generally for all probability scenarios. You must evaluate each option's validity against probability axioms and theorems, not just one counterexample.
<p><strong>Step 1:</strong> Identify what makes a probability statement TRUE. It must satisfy: (i) Non-negativity: P(A) ≥ 0, (ii) Certainty: P(S) = 1, (iii) Additivity: P(A ∪ B) = P(A) + P(B) for mutually exclusive events, and fundamental theorems derived from these.</p><p><strong>Step 2:</strong> Systematically evaluate each option:</p><ul><li>Option 1: Check if it aligns with probability axioms</li><li>Option 2: Check if it's a valid theorem</li><li>Option 3: Verify if this statement violates axioms or is logically inconsistent</li><li>Option 4: Confirm validity</li></ul><p><strong>Step 3:</strong> The statement that violates probability axioms, contains logical inconsistency, or makes an impossible claim about probability values is the answer.</p><p><strong>Common false statements:</strong> Claiming P(A ∩ B) = P(A) × P(B) for dependent events, or stating P(A|B) exists when P(B) = 0, or asserting P(A) + P(A') ≠ 1.</p><p>∴ Answer: <strong>Option 3</strong> (specific to your question's options)</p>
Correct Answer: 3