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Question #41846

Difficulty: 8/10 | 🟢 Free

The Question

Let $E_1, E_2, \dots, E_{81}$ be mutually independent events in a sample space, each with probability $P(E_i) = \frac{1}{3}$ . For each $i \in \{1, 2, \dots, 81\}$ , define the event $F_i = E_i \cup E_{i+1}$ , where $E_{82}$ is defined to be $E_1$ . Let $X$ be the random variable representing the number of events among $F_1, F_2, \dots, F_{81}$ that do not occur. Find the variance of $X$ .

Options

A.
36
B.
28
C.
20
D.
12

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

Variance of Discrete Variable|Indicator Function Logic|De Morgan's Laws in Probability

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to compute the variance of a sum of dependent indicator variables, using De Morgan's laws to correctly identify the event space.

The primary trap lies in either assuming the indicator variables are independent (yielding 20) or forgetting the factor of 2 in the covariance sum (yielding 28).

Additionally, misapplying De Morgan's laws by interpreting 'does not occur' as $E_k^c \cup E_{k+1}^c$ leads to an incorrect covariance structure and a result of 12.

This forces students to rigorously apply $Var(\sum I_k) = \sum Var(I_k) + 2\sum_{i<j} Cov(I_i, I_j)$ while correctly handling the cyclic dependency.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Variance of Discrete Variable
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free