Before You Begin

Question #41810

Difficulty: 8/10 | 🟢 Free

The Question

A particle $P(x,y)$ is placed in the Cartesian plane. A random process selects its domain of uniform distribution: with probability $\frac{2}{5}$ , the particle is distributed uniformly over the region $R_1$ enclosed by the locus $|x| + |y| = 2$ ; with probability $\frac{3}{5}$ , it is distributed uniformly over the region $R_2$ enclosed by the locus $\max(|x|, |y|) = 2$ . Given that the particle's position satisfies the condition $x^2 + y^2 \le 4$ , what is the probability that its domain was $R_1$ ?

Options

A.
$\frac{8}{8 + 3\pi}$
B.
$\frac{50}{89}$
C.
$\frac{4}{4 + 3\pi}$
D.
$\frac{4}{4 + \pi}$

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

Bayes' Theorem Reversal|Conditional Probability Constraints|Locus of a Moving Point

Why This Question Is Tricky

Common Mistakes

This question tests the application of Bayes' Theorem in continuous geometric probability spaces.

The primary trap lies in treating the locus of the moving point as a discrete set of integer coordinates rather than a continuous 2D region.

Students falling for this trap will count the integer coordinates satisfying the conditions (13 points in $R_1$ , 25 points in $R_2$ ) and apply discrete probability to arrive at the incorrect rational option $\frac{50}{89}$ .

The correct approach requires calculating the continuous area of the regions and their intersections with the conditional locus, integrating geometric probability with Bayes' theorem.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Bayes' Theorem Reversal
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free