How do I know what dungeon I'm in?

A few of the old-world dungeons in World of Warcraft that were truly epic places with more content then some of the later raids have been split up in the dungeon finder so it can be done a couple ways. Depending on the particular "mode", it changes who the final boss is, and knowing who it is is important for getting your bag of crap and other bonuses.

My problem is specifically with BRD, one way it ends at the warden ("Detention Block"), the other at the king ("Upper City"). Both start you at the beginning of the instance (unlike Maraudon where it will fling you half-way through it if you just need Princess)

Hilariously, there is no way to determine what dungeon objectives you've been assigned until you complete them.

There is some anecdotal evidence that it's more often than not the lesser of the dungeon objectives (for example, LBRS instead of UBRS), but most likely a form of selection bias is at work there.