The GATE Grind

GATE 2024 CS (CS1) – Question 14

Engineering Mathematics · Probability and Statistics · 1 mark · Multiple choice

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?

  1. The events $X$ and $Y$ are mutually exclusive
  2. The events $X$ and $Y$ are independent
  3. Either event $X$ or $Y$ must occur
  4. Event $X$ is more likely than event $Y$

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Show answer and explanation

Correct answer: (B) The events $X$ and $Y$ are independent

Explanation

P(X)=P(Y)=1/2 and P(X∩Y)=1/4 by symmetry over the relative orders of disjoint pairs, so P(X∩Y)=P(X)P(Y). The events are independent.