Probability
Independent Events
Grade 12
Question:
<p>If \(E\) and \(F\) are independent events such that \(0 < P(E) < 1\) and \(0 < P(F) < 1\) then</p>
<p>(a) \(E, F\) are mutually exclusive</p>
<p>(b) \(E, F\) are independent</p>
<p>(c) \(E, F\) are independent</p>
<p>(d) \(P(E|F) + P(E^c|F) = 1\)</p>
Step-by-Step Solution
Key Concept: For independent events E and F, use P(E∩F) = P(E)·P(F) and P(E∪F) = P(E) + P(F) - P(E)·P(F). The constraint 0 < P(E), P(F) < 1 combined with the relationship between these probabilities determines unique values.
<p><strong>Step 1:</strong> Since E and F are independent events: P(E∩F) = P(E)·P(F)</p><p><strong>Step 2:</strong> For independent events: P(E∪F) = P(E) + P(F) - P(E)·P(F)</p><p><strong>Step 3:</strong> If additional constraints are given (such as P(E∪F) = 3/4, P(E∩F) = 1/4, etc.), substitute into the independence formula to solve for P(E) and P(F)</p><p><strong>Step 4:</strong> Verify that 0 < P(E), P(F) < 1 and that solutions satisfy all given conditions</p><p>∴ Answer: D</p>
Correct Answer: D