Continuity mistake: When Anderton and Agatha approach room "1009" (which looks like "1006"), there's a stairwell/exit to the right of the door. When Anderton bust open "1006" and goes back for Agatha, there's now a hallway to the right of "1009."
Visible crew/equipment: When John Anderton is walking around the hotel room before meeting Leo Crow, just before he spots the photos on the bed, he glances over towards the direction of the bed - if you look to the left of him, you can make out a cameraman's face and hand where he is holding a camera in the reflection of the glass window.
Continuity mistake: The wooden balls etched with the Perpetrator and Victims show the grain running left-to-right across the names when they are first displayed on the cutting machine. But when Anderton removes the balls and places them on the screen (or drops), the names appear to be written with a pen, and the grain is either non-existent or run top-down across the names. This suggests the balls on display in the cutting machine are nicely generated by a computer and much different to the actual props.
Continuity mistake: When the Precrime team are examining the Precogs in the pool, Fletcher folds his arms. In the next shot he folds his arms again.
Continuity mistake: When Lara, standing, is putting on Lamar's tie while he is seated, he gives himself away regarding the drowning of Anne Lively. Lara's arms stop moving and we see her standing straight, her arms by her side, and she says that she never said that Anne was drowned. We cut back to Lamar, and Lara still has her arms around Lamar.
Continuity mistake: When we see the Leo Crow previsions for the first time, we see the numbers "9" and "6" backwards because we are seeing the previsions from behind the screen. But in the cyberparlor we see the previsions from the same side that Anderton and Riley are watching them, and the numbers are still backward.
Answer: Well, there isn't really a simple solution, but here goes. For a pre-vision to form, there have be two things present within the range of the precog ability (which appears to be limited to the Washington area - regardless of the stated plan to take the programme countrywide, there's never any indication that the precogs can sense beyond that range). Firstly, someone with the intent to kill. Secondly, there has to be a target for that intent within the range of the precogs. Anderton is present, and has the intent within him to kill the man who took his son, but has no target - the real kidnapper is presumably either dead or beyond the precog ability. Burgess, by bribing Crow to pretend to be that man, has provided a viable target for Anderton's intent within the range of the precog ability, thus triggering the prevision, and beginning the chain of events.
Tailkinker ★
The above answers the question, but there do appear to be some time travel issues with this plot point in the movie. Burgess set things up for Crow to fake being the kidnapper and thus triggering Jon's desire to kill that person, everything starts by the pre-cogs seeing the future. If the pre-cogs did not exist or did not have the vision, Jon would have never known that Leo Crow existed and would have continued on without having killed anyone. This is unique within the movie, as the other murders would have been commited regardless of whether or not the pre-cogs saw it. In this case, the ONLY reason this murder occurred is because the pre-cogs saw it.
oldbaldyone
Thinking about this a little more, it could be conceivable that Burgess had planned a different option for Jon finding Crow. We just never saw that on screen, because the precogs changed everything to an alternative future timeline once they saw the original murder. Originally, Jon could have been triggered by Burgess himself, stating that they got a lead on his son's murder and pointing him to Crow.
oldbaldyone
No I think Burgess set it up so that Anderton would find Crow because of the precogs, not have a different plan set up before or else it could be possible Burgess himself would be visible in the prevision. He manipulated the system perfectly, he has done it before after all. He knows exactly how the precogs work so he is able to set it up so that it's untraceable. Except, except for the fact there is always a choice. Only then did it go wrong for him. This proves both true for Anderton and Burgess in the end.
lionhead