Consider a permutation sampled uniformly at random from the set of all permutations of \1, 2, 3, ·s, n\ for some n ≥ 4. Let X be the event that 1 occurs before 2 in the permutation, and Y the event that 3 occurs before 4. Which one of the following statements is TRUE?

The events X and Y are independent

The events X and Y are mutually exclusive

The events X and Y are independent

Event X is more likely than event Y

GATE CSE 2024 Session 1 Q14: 1 before 2 and 3 before 4 in a random permutation

Engineering Mathematics
1 mark
MCQ
−1/3 if wrong

Consider a permutation sampled uniformly at random from the set of all permutations of $\{1, 2, 3, \cdots, n\}$ for some $n \ge 4$ . Let $X$ be the event that 1 occurs before 2 in the permutation, and $Y$ the event that 3 occurs before 4. Which one of the following statements is TRUE?

Try it: multiple choice: exactly one correct option

Wrong answer costs 1/3 mark in GATE.

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Answer: B (official GATE 2024 final answer key)

Concept used: Independence: $P(X \cap Y) = P(X)\,P(Y)$ .

First step: By symmetry $P(X) = P(Y) = 1/2$ ; count the relative orders of 1, 2, 3, 4 to find $P(X \cap Y)$ .

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Source: official GATE 2024 Session 1 CS question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key

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GATE CSE 2024 Session 1 Q14: 1 before 2 and 3 before 4 in a random permutation

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