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Question #41715

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ and $B$ be two independent events with probabilities $P(A) = \frac{1}{2}$ and $P(B) = q$ , where $q \in (0, 1)$ . Consider the matrix $M = \begin{pmatrix} P(A \cup B) & P(A^c \cap B^c) & -\frac{1}{2} \\ P(A \cap B) & P(A^c \cup B^c) & -\frac{1}{2} \\ P(A) & P(A^c) & -\frac{1}{2} \end{pmatrix}$ Let $S$ be the set of all values of $q$ for which the eigenvectors of $M$ span $\mathbb{R}^3$ . Then the sum of all elements in $S$ is:

Options

A.
$\frac{1}{2}$
B.
1
C.
$\frac{1}{4}$
D.
S \text{ is an empty set}

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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

Independent vs Disjoint Events|Linear Independence vs Dependence|Even Multiplicity Roots

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of probability and linear algebra, specifically targeting the 'Even Multiplicity' trap.

Students often assume that if a matrix is formed by continuous or distinct probabilistic parameters, its eigenvalues will be distinct, guaranteeing diagonalizability.

Here, the characteristic equation yields a repeated root ( $\lambda = 1/2$ ) regardless of the value of $q$ .

The trap forces students to manually check the nullity of $(M - \lambda I)$ to find the geometric multiplicity.

If they blindly assume diagonalizability due to the variable $q$ , or miscalculate the span of the eigenspace, they will incorrectly conclude that $S$ contains all $q \in (0,1)$ or is empty.

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