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Question #40543

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{1, 2, \dots, 10\}$ . An integer $x$ is chosen uniformly at random from $S$ . Given $x$ , an integer $y$ is chosen uniformly at random from the set $S_x = \{z \in S : |x - z| \le 2\}$ . The probability that $y \ge x$ is

Options

A.
$\frac{371}{600}$
B.
$\frac{27}{44}$
C.
$\frac{3}{5}$
D.
$\frac{67}{120}$

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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

Modulus Inequality Logic|Equiprobable Outcomes|Sample Space Construction

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly construct a sample space under a modulus inequality constraint without falling into the trap of assuming equiprobable outcomes for the restricted set.

A common mistake is to find the total number of pairs $(x,y)$ satisfying $|x-y| \le 2$ (which is 44) and the number of those pairs where $y \ge x$ (which is 27), leading to the incorrect probability of 27/44.

This flawed 'Case Counting' approach ignores the sequential nature of the selection process, which creates a non-uniform probability mass function for the pairs.

Because $y$ is chosen uniformly from the available valid integers after $x$ is chosen, the boundary values of $x$ restrict the sample space of $y$ differently than central values, requiring a re-normalization of the conditional probabilities via the Law of Total Probability.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Modulus Inequality Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free