GATE CS 2024 Set 1 — Question 14
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Engineering Mathematics → Probability & Statistics → Sample Space & Events
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Question
Consider a permutation sampled uniformly at random from the set of all permutations of for some . Let be the event that 1 occurs before 2 in the permutation, and the event that 3 occurs before 4. Which one of the following statements is TRUE?
Correct answer
(B) The events X and Y are independent
Solution
Let be the set of all permutations of the set .
1.Event : 1 occurs before 2. In any random permutation, the relative order of 1 and 2 can only be (1, 2) or (2, 1). By symmetry, both are equally likely. Thus, .
2.Event : 3 occurs before 4. Similarly, the relative order of 3 and 4 can only be (3, 4) or (4, 3), and both are equally likely. Thus, .
3.Independence: To determine if and are independent, we check if .
Consider the relative ordering of the subset . There are possible relative orderings of these four elements, all of which are equally likely in a random permutation of elements.- The event occurs when 1 is before 2 AND 3 is before 4.
- Out of the 24 permutations of , we can count the number of ways where both conditions are met:
- Choose 2 positions out of 4 for the pair in ways. In each choice, there is exactly 1 way to place them such that 1 is before 2.
- The remaining 2 positions must be filled by . There is exactly 1 way to place them such that 3 is before 4.
- Total favorable outcomes = .
- .
- (A) False: , so they are not mutually exclusive.
- (C) False: . Since the probability is not 1, it is not guaranteed that either or must occur.
- (D) False: , so they are equally likely.
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