Probability
Independent Events
Grade 12
Question:
<p>A student appears for tests I, II, and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in tests I, II, and III are, respectively, \(p\), \(q\), and 1/2. If the probability that the student is successful is 1/2, then \(p(1 + q) =\)</p>
<p>(1) 1/2</p>
<p>(2) 1</p>
<p>(3) 3/2</p>
<p>(4) 3/4</p>
Step-by-Step Solution
Key Concept: The student succeeds if he passes (I AND II) OR (I AND III). Set up the probability equation P(success) = P(I∩II) + P(I∩III) - P(I∩II∩III) = 1/2, then use independence to solve for p(1+q).
<p><strong>Step 1:</strong> Define the success event. Student succeeds if: (I ∩ II) ∪ (I ∩ III)</p><p><strong>Step 2:</strong> Using inclusion-exclusion principle (assuming independence):</p><p>P(success) = P(I∩II) + P(I∩III) - P(I∩II∩III)</p><p>= pq + p(1/2) - pq(1/2)</p><p>= pq + p/2 - pq/2</p><p><strong>Step 3:</strong> Factor out p:</p><p>= p(q + 1/2 - q/2)</p><p>= p(q(1 - 1/2) + 1/2)</p><p>= p(q/2 + 1/2)</p><p>= (p/2)(q + 1)</p><p><strong>Step 4:</strong> Set equal to 1/2:</p><p>(p/2)(q + 1) = 1/2</p><p>p(q + 1) = 1</p><p>∴ p(1 + q) = <strong>1</strong></p>
Correct Answer: B