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Question #41714

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The Question

Let $S$ be the sample space of all $2 \times 2$ orthogonal matrices with real entries. A matrix $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$ is chosen uniformly at random from $S$ . (The uniform distribution is defined by choosing the parameter $\theta \in [0, 2\pi)$ uniformly, and the determinant $\det(A) \in \{-1, 1\}$ with equal probability $\frac{1}{2}$ ). Let $E_1$ be the event that $|a_{11} + a_{22}| > \sqrt{2}$ , $E_2$ be the event that $|a_{12} - a_{21}| > \sqrt{2}$ , $E_3$ be the event that $|a_{11}| \ge \frac{1}{2}$ , and $E_4$ be the event that $\det(A) = 1$ . Which of the following statements is correct?

Options

A.
$E_1$ and $E_2$ are independent events, and $P(E_1 \cup E_2) = \frac{1}{2}$
B.
$E_1$ and $E_2$ are mutually exclusive events, and $P(E_1 \cup E_2) = \frac{1}{2}$
C.
$E_3$ and $E_4$ are independent events, and $P(E_3 \cup E_4) = \frac{2}{3}$
D.
$E_1$ and $E_4$ are independent events, and $P(E_1 \cup E_4) = \frac{5}{8}$

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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

Independent vs Disjoint Events|Orthogonal Matrix Conditions|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between independent and mutually exclusive (disjoint) events within a constrained probability space.

The primary trap lies in the psychological assumption that events involving different elements of a matrix (the trace $a_{11}+a_{22}$ versus the skew-trace $a_{12}-a_{21}$ ) behave independently.

However, the orthogonality condition $A^T A = I$ imposes a strict global non-linear coupling on the random variables.

By introducing a modulus function threshold ( $> \sqrt{2}$ ), the problem forces a domain split where satisfying both conditions simultaneously would violate the fundamental identity $\cos^2\theta + \sin^2\theta = 1$ .

Students who fail to parameterize the orthogonal matrix and blindly apply independence formulas will incorrectly choose option A, missing the geometric impossibility of the intersection.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Independent vs Disjoint Events
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free