Probability
Independent Events
Grade 12
Question:
<p><strong>For Problems 10–12:</strong> Let \(S\) and \(T\) are two events defined on a sample space with probabilities \(P(S) = 0.5\), \(P(T) = 0.69\), \(P(S/T) = 0.5\).</p><p>Events \(S\) and \(T\) are</p>
<p>(1) mutually exclusive</p>
<p>(2) independent</p>
<p>(3) mutually exclusive and independent</p>
<p>(4) neither mutually exclusive nor independent</p>
Step-by-Step Solution
Key Concept: Two events are independent if P(S|T) = P(S), and dependent otherwise. Here we check if the conditional probability equals the unconditional probability to determine the relationship.
<p><strong>Step 1:</strong> Recall that events S and T are independent if and only if P(S|T) = P(S).</p><p><strong>Step 2:</strong> Given: P(S) = 0.5 and P(S|T) = 0.5</p><p><strong>Step 3:</strong> Since P(S|T) = 0.5 = P(S), the condition for independence is satisfied.</p><p><strong>Step 4:</strong> We can verify using P(S∩T) = P(S|T)·P(T) = 0.5 × 0.69 = 0.345, and P(S)·P(T) = 0.5 × 0.69 = 0.345. Since P(S∩T) = P(S)·P(T), this confirms independence.</p><p><strong>Step 5:</strong> Events S and T are also not mutually exclusive since P(S∩T) = 0.345 ≠ 0.</p><p>∴ Answer: D (Events S and T are independent)</p>
Correct Answer: D