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Question #40991

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ and $B$ be two events in a sample space such that $P(A) = \frac{1}{2}$ , $P(B) = \frac{1}{3}$ , and $P(A \mid B) = \frac{1}{4}$ . For any event $E$ , let $I_E$ denote its indicator random variable, which equals $1$ if $E$ occurs and $0$ otherwise. Define a random variable $X$ as: $X = 8 I_A + 8 I_B + 8 I_{A^c \cup B^c} - 2024$ Find the variance of $X$ , denoted by $\text{Var}(X)$ .

Options

A.
12
B.
$\frac{316}{9}$
C.
0
D.
$\frac{3}{2}$

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

Variance Invariance under Translation|Indicator Function Logic|De Morgan's Laws in Probability

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to manipulate indicator random variables using De Morgan's Laws and the inclusion-exclusion principle, combined with the translation invariance of variance.

The primary trap lies in assuming that the variance of a sum of indicator variables equals the sum of their variances, which incorrectly presumes independence (leading to 316/9).

Another common pitfall is misapplying De Morgan's Laws directly to the indicator functions (e.g., assuming I_{A^c \cup B^c} = I_{A^c} + I_{B^c}), which falsely reduces X to a constant, yielding a variance of 0.

Finally, students might forget to square the scaling factor when pulling it out of the variance operator, resulting in 3/2.

Success requires algebraically simplifying the random variable into a single indicator function before applying the variance operator.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Variance Invariance under Translation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free