GATE CS 2024 Set 1 Q4 — Probability: Random Variables and Distributions

To determine the relationship between events $X$ and $Y$, we can analyze the relative order of the elements in a randomly chosen permutation of $\{1, 2, 3, \dots, n\}$ for $n \geq 4$:

• Probability of $X$: In any permutation, the element $1$ must appear either before or after $2$. By symmetry, both cases are equally likely. Thus, $P(X) = \frac{1}{2}$.
• Probability of $Y$: Similarly, the element $3$ must appear either before or after $4$. By symmetry, $P(Y) = \frac{1}{2}$.
• Joint Probability $P(X \cap Y)$: Consider the subset of elements $\{1, 2, 3, 4\}$. There are $4! = 24$ equally likely relative orderings of these four elements. For $X \cap Y$ to occur, $1$ must be before $2$, and $3$ must be before $4$.
  • We can choose $2$ positions out of $4$ for $\{1, 2\}$ in $\binom{4}{2} = 6$ ways, placing $1$ in the earlier position and $2$ in the later.
  • The remaining $2$ positions are filled with $3$ and $4$ such that $3$ is in the earlier position and $4$ is in the later ($1$ way).
  Thus, there are exactly $6$ favorable orderings out of $24$, giving $P(X \cap Y) = \frac{6}{24} = \frac{1}{4}$.

Since $P(X \cap Y) = P(X) \cdot P(Y) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$, the events $X$ and $Y$ are independent.