Consider a permutation sampled uniformly at random from the set of all permutations of {1, 2, 3, ⋯ , 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 mutually exclusive
The events 𝑋 and 𝑌 are independent
Either event 𝑋 or 𝑌 must occur
Event 𝑋 is more likely than event 𝑌
Just by reading the question, it is evident that -
Option A: Events X and Y are not mutually exclusive, both events can occur in the same permutation.
Option B: In this case, the occurrence of event 𝑋 indeed does not affect the probability of event 𝑌 occurring, and vice versa. Therefore, Events 𝑋 and 𝑌 are indeed independent.
Option C: It's indeed possible for neither event 𝑋 nor event 𝑌 to occur in a given permutation. For example, in the permutation 2143, neither 1 occurs before 2 nor 3 occurs before 4.
Option D: Both events are symmetric, their probabilities are the same.
The correct answer is B.