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Question #41822

Difficulty: 8/10 | 🟢 Free

The Question

An urn contains $4$ red balls and $6$ black balls. Let $X$ be the number of red balls obtained when $3$ balls are drawn without replacement. Let $Y$ be the number of red balls obtained when $3$ balls are drawn with replacement. Define the function $g(t) = 15 E[|X - t|] + 125 E[|Y - t|]$ for $t \in \mathbb{R}$ . Find the minimum value of $g(t)$ .

Options

A.
87
B.
95
C.
116
D.
129

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

Sampling With/Without Replacement|Random Variable Expectation|Critical Point vs Stationary Point|Non-Differentiable Extrema

Why This Question Is Tricky

Common Mistakes

This problem tests the optimization of the expectation of an absolute deviation function, $E[|Z - t|]$ .

The core trap lies in the non-differentiability of this function at the points in the support of the distribution.

Students accustomed to finding extrema by setting the derivative to zero ( $g'(t) = 0$ ) will fail because the derivative is a step function that never equals zero; it jumps across zero at the median.

This forces the student to recognize that the minimum occurs at a critical point where the derivative is undefined (a corner point), specifically the median of the distribution, rather than the mean (which is a common incorrect assumption for absolute deviation).

Furthermore, it rigorously tests the calculation of probabilities for both sampling with and without replacement.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Sampling With/Without Replacement
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free