GATE CS 2024 Set 1 Q.14: Probability and Statistics

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?

The events $X$ and $Y$ are mutually exclusive

The events $X$ and $Y$ are independent

Either event $X$ or $Y$ must occur

Event $X$ is more likely than event $Y$

Official answer and solution

Official answer: B
From: Official GATE 2024 answer key

Solution

Use symmetry of disjoint element pairs in a uniform random permutation to compute $P(X), P(Y),$ and $P(X \cap Y)$ .

By symmetry in a uniformly chosen permutation of $\{1, 2, \dots, n\}$ :

$1$ and $2$ are equally likely to appear in either relative order, so $P(X) = \frac{1}{2}$ .
$3$ and $4$ are equally likely to appear in either relative order, so $P(Y) = \frac{1}{2}$ .
Since $\{1, 2\}$ and $\{3, 4\}$ are disjoint pairs of elements ( $n \ge 4$ ), all $2 \times 2 = 4$ relative orderings of $(1, 2)$ and $(3, 4)$ partition the $n!$ permutations into 4 equal-sized classes of size $n!/4$ . Thus:

P(X \cap Y) = \frac{1}{4} = P(X)P(Y)

Hence events $X$ and $Y$ are independent.

Final answer

Option (B)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2024 Computer Science and Information Technology - Set 1 (CS1)
Question: 14 of 65
Section: Computer Science and Information Technology

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