Probability
Conditional Probability
Grade 12
Question:
<p>A student appears for tests I, II and III. The student is successful if he passes in tests I, II or I, III. The probabilities of the student passing in tests I, II and III are respectively \(p\), \(q\) and \(\frac{1}{2}\). If the probability of the student to be successful is \(\frac{1}{2}\) then</p>
<p>(a) \(p(1+q) = 1\)</p>
<p>(b) \(q(1+p) = 1\)</p>
<p>(c) \(pq = 1\)</p>
<p>(d) \(p + \frac{1}{q} + 1 = 1\)</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, assuming independence, then solve for the relationship between p and q.
<p><strong>Step 1:</strong> Identify success condition. Student is successful if: (passes I AND II) OR (passes I AND III).</p><p><strong>Step 2:</strong> Express using probability notation. Let events be independent.</p><p>P(success) = P[(I∩II) ∪ (I∩III)]</p><p><strong>Step 3:</strong> Use inclusion-exclusion principle.</p><p>P[(I∩II) ∪ (I∩III)] = P(I∩II) + P(I∩III) - P(I∩II∩III)</p><p>= pq + p(1/2) - p·q·(1/2)</p><p>= pq + p/2 - pq/2</p><p><strong>Step 4:</strong> Set equal to 1/2.</p><p>pq + p/2 - pq/2 = 1/2</p><p><strong>Step 5:</strong> Simplify.</p><p>pq(1 - 1/2) + p/2 = 1/2</p><p>pq/2 + p/2 = 1/2</p><p>p(q/2 + 1/2) = 1/2</p><p>p(q + 1)/2 = 1/2</p><p><strong>Step 6:</strong> Final relationship.</p><p>p(q + 1) = 1</p><p>∴ Answer: A (relationship is p(1+q) = 1 or equivalent)</p>
Correct Answer: A