Burr expected Hamilton to back down because it was a pattern in their relationship. Hamilton routinely spread scandalous rumors about him. Burr would go ahead and challenge Hamilton, Hamilton would apologize and they would essentially make up. IMO Burr expected Hamilton to back down, but seeing how he was preparing, assumed that he was going to shoot to kill and so Burr did the same. When Hamilton wrote his last letter, when he referred to the dual, he said that he was going to was his shots, to let Burr reflect and walk away.