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Question #41712

Difficulty: 8/10 | 🟢 Free

The Question

Let $\hat{a}, \hat{b}, \hat{c}$ be three mutually perpendicular unit vectors in $\mathbb{R}^3$ forming a right-handed coordinate system. Let $\hat{x}$ be a random unit vector chosen uniformly from the surface of the unit sphere centered at the origin. Let $V$ be the volume of the parallelepiped whose coterminous edges are the vectors $\hat{a}-\hat{x}, \hat{b}-\hat{x}$ , and $\hat{c}-\hat{x}$ .

Define two events based on this volume: $E_1$ : $V \le 2$ $E_2$ : $V \ge 1$

Find the conditional probability $P(E_2 \mid E_1)$ .

Options

A.
$\frac{\sqrt{3}-1}{2}$
B.
$\frac{1}{2}$
C.
$\frac{\sqrt{3}}{3}$
D.
$\frac{3-\sqrt{3}}{2}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Conditional Probability Constraints|Determinant as Geometric Area/Volume|Position Vector Origin Dependence|Unit Vector Normalization

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle stochastic vectors and determinant properties.

The primary trap is 'Position Vector Origin Dependence': students often incorrectly assume that the volume of a parallelepiped formed by vectors $\hat{a}-\hat{x}, \hat{b}-\hat{x}, \hat{c}-\hat{x}$ is invariant to the shift $\hat{x}$ (yielding $V=1$ ), confusing it with the volume formed by position vectors of vertices.

The secondary trap is 'Unit Vector Normalization': when expanding the scalar triple product, students must recognize that the vector $\hat{a}+\hat{b}+\hat{c}$ has a magnitude of $\sqrt{3}$ , not $1$ .

Failing to normalize this vector before applying the uniform distribution of the dot product on a sphere leads to incorrect interval bounds and a flawed probability of $1/2$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Conditional Probability Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free