ProbabilityMCQPYQ June 23Question 3334 of 187
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For any two events 'A' and 'B' it is known that P(A)=2/3\displaystyle P(A) = 2/3, P(B)=3/8\displaystyle P(B) = 3/8 and P(A∩B)=1/4\displaystyle P(A \cap B) = 1/4, then the events A and B are:

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option b — Mutually not exclusive and Independent

All Options:

  • AMutually exclusive and Independent
  • BMutually not exclusive and Independent
  • CMutually exclusive but not Independent
  • DNeither independent nor mutually exclusive

Detailed Solution & Explanation

**Analyzing the Relationship Between Events A\displaystyle A and B\displaystyle B** Given: - P(A)=2/3\displaystyle P(A) = 2/3 - P(B)=3/8\displaystyle P(B) = 3/8 - P(A∩B)=1/4\displaystyle P(A \cap B) = 1/4 **1. Mutually Exclusive Check:** Two events are mutually exclusive if they cannot occur at the same time, meaning: P(A∩B)=0P(A \cap B) = 0 Since P(A∩B)=1/4≠0\displaystyle P(A \cap B) = 1/4 ≠\displaystyle \neq 0, the events are **not mutually exclusive** (mutually not exclusive). **2. Independence Check:** Two events are independent if the probability of their joint occurrence is equal to the product of their individual probabilities: P(A∩B)=P(A)×P(B)P(A \cap B) = P(A) ×\displaystyle \times P(B) Let's compute the product: P(A)×P(B)=23×38=624=14P(A) ×\displaystyle \times P(B) = 23\displaystyle \frac{2}{3} ×\displaystyle \times 38=624=14\displaystyle \frac{3}{8} = \frac{6}{24} = \frac{1}{4} Since P(A∩B)=1/4\displaystyle P(A \cap B) = 1/4, the condition for independence holds. Thus, the events are **independent**. Therefore, the events A\displaystyle A and B\displaystyle B are mutually not exclusive and independent. 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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