This special random subset of uniformly distributed numbers is still uniformly distributed? I asked similar question in A special random subset of uniformly distributed numbers is still uniformly distributed?
Here, I slightly change my random number generation method, and want to see whether the sampled numbers are still uniformly distributed.
Assume that I have a value range [1,1000].
Goal: I want to have 10 numbers randomly sampled from [1,1000].
case1:
I sample 20 numbers, a1, ..., a20 from [1,1000].

Then I sample b1, ..., b10 from [a1, a2, ..., a20].

b1, ..., b10 are what I want.

case2:
I sample 20 numbers, a1, ..., a20 from [1,1000].

Then I sort a1, ..., a20.

For notation simplicity, I assume that after sorting, a1< ...< a20.

Then, for each consecutive pair of numbers, 

I uniformly at random select one number.

Eventually, I can get 10 numbers as well.

I know the numbers in case1 are uniformly distributed.
Does anyone have idea whether the numbers in case2 are uniformly distributed?

Let us do a much simplified calculation. We want to choose $2$ numbers out of the integers $1$ to $10$. In Scheme 1, we choose $4$ numbers, then choose $2$ of them. In Scheme 2, we choose $4$, order them, pick one from the first group of two, and one from the second group.
Suppose that we are choosing the $4$ without replacement. With Scheme 1, the probability of ending up with the set $\{9,10\}$ is $\frac{1}{45}$. With Scheme 2, it is $0$.
With replacement gets more complicated, because even Scheme 1 bifurcates. But in various explicit calculations, I get different probabilities. 
Remark: There have been several related posts. In all of them, the version of Scheme 2 that is used produces a tamer, more "evenly" distributed result than true random sampling.
Reminds me of an anecdote, perhaps not true, about when bridge tournaments switched from manually shuffled decks (then replicated for the other table), to hands generated by a random number generator. Veteran bridge players complained that the computer was dealing "wilder" than normal hands, for example hands with more voids. They were sort of right: manually shuffled decks produce hands not uniformly distributed over all possible hands. 
