Bellatrix truly believed she was doomed, that's a strong belief against harrys position. Harry himself didn't fully believe it either. I'd argue that the extent of bellatrixes disbelief was greater then the great halls, although it's hard to say. That coupled with the fact that harry wasn't a true believer leads to a consensus that they're not leaving. Now we have 2 true believer on harrys side and a bunch of people who probably don't have real strong opinions one way or the other.
Also, it's entirely possible that magical ability decides how many "votes" you get in the consensus!dementors actions. Albus has reason to very very strongly believe that the dementor will do as he's told. Whether that's enough to overwhelm the rest of the voters, who believe that the dementor will not do what harry says, but fear them and would certainly be thrown off gourd by the first question, is hard to say.
My hope is that is answers "yes" to the first question, enough to raise enough doubt that the second question can also be a yes. Once the second question has been answered in the mysterious yes manner, hopefully it confuses the expectations of the crowd enough to allow the dementor to be ordered about. You start with a question that can be answered innocuously enough, but changes the expectations enough. Then you keep doing that. Like murder gandhi.
Why does Albus have such a belief? He witnessed Harry kill a dementor, yes, but not control one.
That's not to say that he would disbelieve that Harry could do it, but I see no reason for him to have a reason to "very very strongly believe" it.
EDIT: New discussion thread here.
This is a new thread to discuss Eliezer Yudkowsky's Harry Potter and the Methods of Rationality and anything related to it. With two chapters recently the previous thread has very quickly reached 500 comments. The latest chapter as of 17th March 2012 is Ch. 79.
There is now a site dedicated to the story at hpmor.com, which is now the place to go to find the authors notes and all sorts of other goodies. AdeleneDawner has kept an archive of Author's Notes. (This goes up to the notes for chapter 76, and is now not updating. The authors notes from chapter 77 onwards are on hpmor.com.)
The first 5 discussion threads are on the main page under the harry_potter tag. Threads 6 and on (including this one) are in the discussion section using its separate tag system. Also: one, two, three, four, five, six, seven, eight, nine, ten.
As a reminder, it's often useful to start your comment by indicating which chapter you are commenting on.
Spoiler Warning: this thread is full of spoilers. With few exceptions, spoilers for MOR and canon are fair game to post, without warning or rot13. More specifically: