Recently A problem in this site asked: shuffling a class of students sitting in pairs - what is the probability that at least one pair "survives". I was reading this answer by leonbloy, which provides very elegant way for approximating (for a large number of students) a rather computationally-difficult probability question. The answer states that: "We can expect that the events $E_i \equiv$ "the pair $i$ survived", are asymptotically independent".
I can clearly see that any given pair of events like $E_z$, $E_y$ is asymptotically independent; but initially, it does not seem immediately trivial to me that they are asymptotically fully/manually independent in the sense that would allow the approximation to work. In fact, strictly speaking, they are not fully independent since the "last" pair is determined by all the previous ones: though I understand this is marginal case and should not have an impact on the result. But what if we fix half of the pairs: can we say the second half is not "influenced"/dependent by this? Well, I guess we can at infinity - but how to show it technically? How to show that we can use this approximation method?
