My first thought was that the mantra of “shut up and calculate” clearly means that we shouldn’t get angry at the administrator for doing that. But Harry’s conversations with Dumbledore seemed to go the opposite way. Dumbledore was trying to calculate how to do the most good even if some of the kids get hurt, and Harry was getting angry at him for it.
My guess now is that Harry’s not angry at the administrator for calculating. He’s angry at the administrator for not calculating how to parley a million dollars into a kidney and a bunch of equipment, salaries, et cetera, and a breakthrough cancer treatment as long as you’re spending money, anyway. And that, I presume, is what the “Cheating” subtitle on the prelude means.
It suddenly occurs to me that Dumbledore has seen two interactions between Harry and a Dementor. In the first one, it almost destroys him. In the second, he casts a Patronus that destroys it. Neither would seem to provide the kind of evidence that you would need to confidently assume that other Dementors would run away from you if you said “Boo” to them.
So, is this enough evidence for Dumbledore to decide that he’s wrong about who broke Bellatrix Black out of Azkaban?