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Question #41723

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a random variable representing the number of successes in $25$ independent Bernoulli trials, each with probability of success $p \in (0, 1)$ . If the coefficient of variation of $X$ is at most $\frac{\sqrt{2}}{10}$ , then the maximum possible standard deviation of $X$ is:

Options

A.
$\frac{5\sqrt{2}}{3}$
B.
$\frac{5}{2}$
C.
$\frac{50}{9}$
D.
$\frac{25}{4}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Bernoulli Trials Logic|Standard Deviation and Coefficient of Variation|Maximum/Minimum of Quadratic Expressions

Why This Question Is Tricky

Common Mistakes

This question tests the optimization of a quadratic function representing the variance of a binomial distribution under a restricted domain.

The primary trap lies in blindly applying the vertex formula for the maximum of $p(1-p)$ at $p=0.5$ .

By constraining the coefficient of variation, the domain of $p$ is restricted to $[2/3, 1)$ , forcing the maximum variance onto the boundary $p=2/3$ .

Students who ignore the domain constraint will incorrectly choose the unconstrained maximum standard deviation of $5/2$ , while those who confuse variance with standard deviation will fall for the other distractors.

Quick Facts About This Question

Question Information

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