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Question #41089

Difficulty: 8/10 | 🟢 Free

The Question

Let $A_1, A_2, A_3, A_4$ be four mutually independent events with probabilities $P(A_k) = \frac{1}{k+1}$ for $k=1, 2, 3, 4$ . Let $M$ be a $4 \times 4$ matrix whose entries are defined by $M_{ij} = P(A_i \cup A_j) - P(A_i \cap A_j)$ . Find the value of $\det(M)$ .

Options

A.
$-\frac{101}{900}$
B.
0
C.
$\frac{101}{900}$
D.
$-\frac{101}{225}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Inclusion-Exclusion Principle|Row/Column Exchange Parity|Transitivity of Identity

Why This Question Is Tricky

Common Mistakes

The question tests the intersection of probability (Inclusion-Exclusion) and linear algebra.

The primary trap is the 'Transitivity of Identity' assumption: students correctly identify the off-diagonal entries as $p_i q_j + p_j q_i$ but incorrectly assume this algebraic identity holds transitively for the diagonal elements.

If they fail to subtract the diagonal correction matrix $2D$ , they evaluate the determinant of a pure rank-2 matrix ( $u v^T + v u^T$ ), which is $0$ for $n \ge 3$ , leading them straight to the trap Option B.

Correcting the diagonal requires recognizing the matrix as a rank-2 update to a diagonal matrix and applying the Matrix Determinant Lemma.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Inclusion-Exclusion Principle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free