Before You Begin

Question #41802

Difficulty: 8/10 | 🟢 Free

The Question

A line $L$ with direction vector $\vec{d}$ is generated such that $\vec{d}$ is a random vector uniformly distributed on the surface of the unit sphere in $\mathbb{R}^3$ . Let $P$ be a fixed plane in $\mathbb{R}^3$ . Let $\theta \in \left[0, \frac{\pi}{2}\right]$ be the acute angle between the line $L$ and the plane $P$ . If $E_1$ is the event that $\theta \ge \frac{\pi}{6}$ and $E_2$ is the event that $\theta \le \frac{\pi}{3}$ , then the conditional probability $P(E_2 \mid E_1)$ is equal to

Options

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

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

Conditional Probability Constraints|Angle between line and plane|Line-Plane intersection logic

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of 3D vector geometry and continuous probability distributions.

The primary trap lies in the probability density function (PDF) of the angle.

Students often memorize that the angle $\phi$ between a random 3D vector and a fixed axis has a sine distribution ( $f(\phi) = \sin\phi$ ).

When asked for the angle between a line and a plane, they incorrectly apply this same sine PDF.

However, the angle to the plane is the complement of the angle to the normal ( $\theta = \pi/2 - \phi$ ), which transforms the PDF from sine to cosine ( $g(\theta) = \cos\theta$ ).

Using the incorrect sine PDF leads directly to the trap option $1 - 1/\sqrt{3}$ , while assuming a uniform distribution leads to $1/2$ .

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