Agatha intentionally created the Leo Crow previsionHonestly, this is the only thing that makes the movie make sense. Agatha knew that Burgess had drowned her mother, Anne Lively, but couldn't prove it. So, she manipulated her own ability and created a vision that she knew would result in Anderton finding and uncovering the person responsible. She gave Burgess some foreknowledge, but ultimately the only thing that sent Anderton on his crusade was the fact that he had seen Agatha's prevision. Therefore, Agatha herself created the vision, and she had the ability to know what Anderton would do when he saw it.
There are other Precogs out thereAnne only came for Agatha, implying the Dash and Arthur aren't biologically related to her. This would imply that neuroin could have similar effects on any pregnancy, and more than just the three exist. It wouldn't even be that hard to evade the authorities. Surely there were some that were born outside of the jurisdiction. Some may have been hidden away by a non-addict parent that did care when their abilities emerged. Any others likely would have hidden their abilities themselves if they were born after Pre-crime's creation, fearing the same captivity that the main three underwent. The tv show may even look into this.
- Would it have been practical for Precrime to expand from D.C. to the entire country, without a larger number of precogs? In the car after Anderton abducts Agatha, she says, "I'm tired of the future," which suggests the Precogs were overworked already.
Burgess had Sean killed.And he did it so that he could create the perfect face of the PreCog program - John Anderton, the cop who lost someone he loved, and was genuinely motivated to prevent such a thing from happening again.