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Question #40315

Difficulty: 8/10 | 🟢 Free

The Question

Let $A, B,$ and $C$ be three exhaustive events in a sample space such that $P(A) = \frac{1}{2}$ , $P(B) = \frac{2}{3}$ , $P(C) = \frac{3}{4}$ , and $P(A \cap B \cap C) = \frac{1}{6}$ . Find the probability of the event $E = (A^c \cup B^c) \cap (B^c \cup C^c) \cap (C^c \cup A^c)$ .

Options

A.
$\frac{1}{4}$
B.
$\frac{1}{8}$
C.
$\frac{5}{12}$
D.
$\frac{1}{3}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Set Identity Laws|De Morgan's Laws in Probability|Exhaustive Events Concept|Mutually Exclusive vs Independent

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize Set Identity laws (De Morgan's) with probability axioms, specifically targeting the 'Exhaustive Events' constraint.

A common trap is reflexively assuming independence when individual probabilities are given, leading to the incorrect calculation of pairwise intersections (e.g., P(A ∩ B) = P(A)P(B)) and yielding the trap answer 1/8.

The exhaustive condition (P(A ∪ B ∪ C) = 1) forces a dependency among the events, requiring the student to extract the exact sum of pairwise intersections via the inclusion-exclusion principle before evaluating the simplified union.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Set Identity Laws
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free