A Probability distributions in an ordered set of extracted elements

AI Thread Summary
The discussion centers on determining the probability distribution function for the i-th position of an ordered set of n elements extracted from a set S with a total order relation R. The original poster seeks existing solutions or literature on this specific probability problem, noting difficulty in finding relevant resources. A suggestion is made to approach the problem by counting the combinations of elements smaller and larger than the x-th element in the ordered set. Participants encourage exploring the problem further, indicating that it may not be overly complex. The conversation emphasizes the need for mathematical exploration and potential solutions in combinatorial probability.
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Finding the probability distributions in an ordered set of random extracted elements
Hello, I tried looking for an existing solution for the following problem:
"Assume that S is a set of d elements, and R is a total order relation on S. Assume that n elements are randomly extracted from S, and then they are ordered according to R. Which is the probability that in the i-th position of the ordered set of n elements there is the x-th element of S? (where the x-th element of S is intended according to the R).

In other words, which is the probability distribution function for the i-th position of the ordered set of n elements? Do you know whether such problem has already been solved? I did not find any specific solution in literature or elsewhere. I tried do find my own solution, but I would like to know if something already exist.
 
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I don't think this is that hard. To get the xth element of R in the ith position, you need to pick x-1 elements smaller than it, and n-x-1 elements larger than it. You can just count the number of ways to do this. Why don't you give it a shot?
 
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