<p>\(AB\) and \(\overline{AB}\) are mutually exclusive and exhaustive events. If \(P(AB) = \frac{1}{25}\), then \(P(\overline{A} \cdot \overline{B}) = 1 - P(AB)\) equals:</p>
Step-by-Step Solution
Key Concept: When AB and Ā·B̄ are mutually exclusive and exhaustive, they partition the sample space. Therefore P(AB) + P(Ā·B̄) = 1, making P(Ā·B̄) = 1 - P(AB) by definition, not by probability rules.
<p><strong>Step 1:</strong> Recognize what 'mutually exclusive and exhaustive' means. Two events are mutually exclusive if they cannot occur together: P(AB ∩ Ā·B̄) = 0. They are exhaustive if they cover all outcomes: AB ∪ Ā·B̄ = S (sample space).</p><p><strong>Step 2:</strong> For any partition of the sample space, the sum of probabilities equals 1: P(AB) + P(Ā·B̄) = 1</p><p><strong>Step 3:</strong> Therefore: P(Ā·B̄) = 1 - P(AB) = 1 - 1/25 = 24/25</p><p>∴ Answer: <strong>24/25</strong> (Option B)</p>
Correct Answer: B