Probability
Independent Events
Grade 12
Question:
<p>Three persons P, Q and R independently try to hit a target. If the probabilities of their hitting the target are \(3/4\), \(1/2\) and \(5/8\), respectively, then the probability that the target is hit by P or Q but not by R is:</p>
<p>\(\dfrac{39}{64}\)</p>
<p>\(\dfrac{21}{64}\)</p>
<p>\(\dfrac{15}{64}\)</p>
<p>\(\dfrac{9}{64}\)</p>
Step-by-Step Solution
Key Concept: Since the events are independent, use P(A and B and C) = P(A)·P(B)·P(C). The phrase 'P or Q but not R' means (P hits OR Q hits) AND R misses, which expands as P(P∪Q)·P(R').
<p><strong>Step 1:</strong> Identify what 'P or Q but not R' means: (P hits OR Q hits) AND R doesn't hit.</p><p><strong>Step 2:</strong> Calculate P(R doesn't hit) = P(R') = 1 - 5/8 = 3/8</p><p><strong>Step 3:</strong> Calculate P(P or Q) using the addition rule for independent events:<br/>P(P∪Q) = P(P) + P(Q) - P(P)·P(Q)<br/>= 3/4 + 1/2 - (3/4)(1/2)<br/>= 3/4 + 1/2 - 3/8<br/>= 6/8 + 4/8 - 3/8 = 7/8</p><p><strong>Step 4:</strong> Since hitting by (P or Q) and not hitting by R are independent events:<br/>P[(P∪Q)∩R'] = P(P∪Q)·P(R')<br/>= (7/8)·(3/8) = 21/64</p><p>∴ Answer: B (21/64)</p>
Correct Answer: B