ProbabilityMCQPYQ Dec 23Question 3336 of 187
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If P(A)=1/2\displaystyle P(A) = 1/2 and P(B)=1/3\displaystyle P(B) = 1/3 and P(A∩B)=2/3\displaystyle P(A \cap B) = 2/3 then find P(A∩B)\displaystyle P(A \cap B)

Options

A1/4\displaystyle 1/4
B2/3\displaystyle 2/3
C1\displaystyle 1
D1/2\displaystyle 1/2
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Correct Answer

✅ Option b — 2/3\displaystyle 2/3

All Options:

  • A1/4\displaystyle 1/4
  • B2/3\displaystyle 2/3
  • C1\displaystyle 1
  • D1/2\displaystyle 1/2

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Detailed Solution & Explanation

**Evaluating Probability Identity with Typos** Given: - P(A)=1/2\displaystyle P(A) = 1/2 - P(B)=1/3\displaystyle P(B) = 1/3 - P(A∩B)=2/3\displaystyle P(A \cap B) = 2/3 We are asked to find P(A∩B)\displaystyle P(A \cap B). **Step 1: Literal Interpretation** From the given information in the question text, the value of P(A∩B)\displaystyle P(A \cap B) is directly given as 2/3\displaystyle 2/3. P(A∩B)=23P(A \cap B) = \frac{2}{3} This matches Option B. **Step 2: Analysis of Typos** Mathematically, P(A∩B)≤P(A)=1/2\displaystyle P(A \cap B) \le P(A) = 1/2. Thus, the given value P(A∩B)=2/3\displaystyle P(A \cap B) = 2/3 is inconsistent with probability axioms. If we assume the third given value was meant to be P(A∪B)=2/3\displaystyle P(A \cup B) = 2/3, then using the Addition Theorem: P(A∩B)=P(A)+P(B)−P(A∪B)=12+13−23=16P(A \cap B) = P(A) + P(B) - P(A \cup B) = \frac{1}{2} + \frac{1}{3} - \frac{2}{3} = \frac{1}{6} Since 1/6\displaystyle 1/6 is not in the options, and the official answer key lists Option C (1\displaystyle 1), this indicates a layout/printing error in the original exam paper. If we strictly follow the given question text asking for the value of the given quantity P(A∩B)=2/3\displaystyle P(A \cap B) = 2/3, it corresponds to Option B. Hence, **Option B** is the correct answer.

About This Chapter: Probability

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Probability Operations, Expected Value

A logic-heavy chapter dealing with random experiments, events (mutually exclusive, exhaustive), set theory probability, conditional probability, and Bayes' Theorem. It forms the basis for Theoretical Distributions.

View Official ICAI Syllabus

Exam Strategy Tip

Always draw a quick Venn Diagram or tree when faced with 'At least one' or 'Only A but not B' wording. It saves you from double-counting.

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