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Question #40646

Difficulty: 8/10 | 🟢 Free

The Question

A biased 3-sided die with faces labeled $A, B,$ and $C$ is rolled 6 times. The probabilities of landing on $A, B,$ and $C$ on any given roll are $\frac{1}{2}, \frac{1}{4},$ and $\frac{1}{4}$ respectively. Let $E$ be the event that every face appears at least once during the 6 rolls. If $X$ is the random variable representing the number of times face $A$ appears, find the conditional expectation $E[X \mid E]$ .

Options

A.
$\frac{57}{22}$
B.
3
C.
$\frac{380}{143}$
D.
$\frac{5}{2}$

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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

Multinomial Theorem|Random Variable Expectation|Complementary Counting

Why This Question Is Tricky

Common Mistakes

This question tests the calculation of conditional expectation within a multinomial distribution framework.

The primary trap is the 'Complementary Counting' hurdle: students often miscalculate the sample space $P(E)$ by treating the constraints as independent binomial events, or they attempt to find the conditional distribution of $X$ directly, leading to intractable algebraic sums.

The elegant, JEE-Advanced level bypass requires using the linearity of expectation with indicator variables ( $E[X \cdot \mathbb{I}_E]$ ).

Recognizing that $E[X \cdot \mathbb{I}_{B=0}] = E[X|B=0]P(B=0)$ and that terms where $A=0$ trivially zero out the expectation is the key to avoiding the lethal collision of dependent variable states.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Multinomial Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free