Question

Consider a permutation sampled uniformly at random from the set of all permutations of \( \{1, 2, 3, \dots, n\} \) for some \( n \geq 4 \). 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?

• A. The events \( X \) and \( Y \) are mutually exclusive.
• B. The events \( X \) and \( Y \) are independent.
• C. Either event \( X \) or \( Y \) must occur.
• D. Event \( X \) is more likely than event \( Y \).

Hint:

Independence of events means the outcome of one event does not influence the probability of the other.

Answer 1

The Correct Option is B

Step 1: Define the events \( X \) and \( Y \).
Event \( X \): 1 occurs before 2 in the permutation.
Event \( Y \): 3 occurs before 4 in the permutation. Step 2: Determine probabilities.
For any two distinct elements in a permutation, the probability of one occurring before the other is: \[ P(X) = P(Y) = \frac{1}{2}. \] Step 3: Check independence.
Events \( X \) and \( Y \) are independent if: \[ P(X \cap Y) = P(X) \cdot P(Y). \]
Since the order of 1 and 2 is independent of the order of 3 and 4, we have: \[ P(X \cap Y) = \frac{1}{4}, \quad P(X) \cdot P(Y) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}. \] Step 4: Conclusion. Events \( X \) and \( Y \) are independent as the probabilities match. Final Answer: \[ \boxed{\text{The events } X \text{ and } Y \text{ are independent.}} \]