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Question #41060

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ and $B$ be two events with probabilities $p$ and $q$ respectively, where $0 < p, q < 1$ . Let $I_A$ and $I_B$ be their corresponding indicator functions. If $E[(I_A - I_B)^2] = p + q$ and $\frac{E[I_A I_B] - 2pq}{p-1} \ge q(2p+1)$ , then which of the following represents the correct range of values for $p$ ?

Options

A.
$\left[\frac{1}{2}, 1\right)$
B.
$\left(0, \frac{1}{2}\right]$
C.
$\left[\frac{1}{\sqrt{2}}, 1\right)$
D.
$\left(0, \frac{1}{\sqrt{2}}\right]$

Check Your Answer

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

Mutually Exclusive vs Independent|Indicator Function Logic|Sign Flipping Rule

Why This Question Is Tricky

Common Mistakes

This question tests the deduction of event relationships via indicator functions and the rigorous manipulation of inequalities with probability parameters.

The first layer requires students to recognize that $E[(I_A - I_B)^2] = p + q$ implies $P(A \cap B) = 0$ , meaning the events are mutually exclusive, not independent.

The lethal trap lies in the 'Sign Flipping Rule' during the algebraic manipulation of $\frac{-2pq}{p-1} \ge q(2p+1)$ .

Because $p$ is a probability strictly less than 1, the denominator $(p-1)$ is negative.

Students who mechanically cross-multiply without reversing the inequality sign will arrive at $p \le 1/2$ , falling directly into the trap option $(0, 1/2]$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Mutually Exclusive vs Independent
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free