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Question #41063

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{1, 2, 3, 4, 5, 6\}$ . An ordered pair of subsets $(A, B)$ of $S$ is chosen uniformly at random from all such pairs that satisfy $A \cup B = S$ . If it is given that the set difference $A \setminus B$ contains exactly one element, what is the probability that the symmetric difference $A \Delta B$ contains exactly three elements?

Options

A.
$\frac{5}{16}$
B.
$\frac{80}{243}$
C.
$\frac{1}{6}$
D.
$\frac{5}{32}$

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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

Equiprobability Fallacy|Total Selection (at least one)|Difference vs. Symmetric Difference

Why This Question Is Tricky

Common Mistakes

The question tests the Equiprobability Fallacy and independence assumptions in combinatorial set operations.

Students often mistakenly assume that the sizes of disjoint regions like $A \setminus B$ and $B \setminus A$ are independent, leading to the trap answer 80/243 (calculating the unconditional probability of $|B \setminus A| = 2$ ).

Furthermore, the equiprobability trap (1/6) catches students who assume the remaining elements form subsets of all possible sizes with equal likelihood, ignoring the underlying binomial distribution of the elements.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Equiprobability Fallacy
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free