Consider the following exercise from Cover and Thomas:
And the given solution from the solutions manual:
It is reasonably clear that these bounds are valid (one simply follows the counting argument given in the chapter, but applied to triple X,Y,Z instead of the pair X,Y). It is not clear to me, however, that these are the tightest bounds that can be found. In particular, it is immediately apparent that the pairwise typicality condition is not used in this solution. This seems strange, to me - does the inclusion of the pairwise typicality condition not provide us any information we can use to obtain narrower bounds? If not, why not - and would that imply that the inclusion of the pairwise typicality condition is in some sense unnecessary or redundant?
The pairwise conditions are "not useful" merely in the sense that the problem is not asking about a situation that they control. The problem asks us only about the case where the variables are drawn totally independently. The pairwise conditions control other, similar scenarios - such as when X and Y are drawn together, but (jointly) independently of Z.