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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 be 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

Correct answer: The events X and Y are independent

Solution

Events X and Y are independent because the relative positions of 1 and 2 do not affect the relative positions of 3 and 4 in a random permutation. Each pair's ordering is determined independently, leading to a probability of 1/2 for each event occurring.

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