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Originally Posted by asdfasdf32
I'm asserting two things (1) it's more likely than not for a runaway to be outside of the prerequisite ~1 square mile surrounding their house if the police are already searching for a body
I have no idea what is making you think there's a radius that creates the transition between a search for a body and a search for a live person. The "searching for a body" (as far as I know) is not bounded by physical distance. It could be tied to the particular details of the written note, as well as the circumstances within the family. However, it's most likely tied to the amount of time that has elapsed since the discovery.
OP stated that it's the end of a 3 day retreat, so the child has been potentially missing for four days. (I somehow recall that it's 24 hours before a missing person report can be filed -- although this has changed with the implementation of the Amber Alert system, but this is only applied to children, and even then I don't think it's a formal missing person report. But either way, if the police are involved, it's not as if the child has been gone for an afternoon.)
As to the location, in an area with a lot of cover (cornfields, forest), it's easy to hide if you're trying to hide. You don't need to go very far to become invisible. Most search and rescue operations operate with the assumption that the lost person wants to be found, so there's a lot of calling out and that sort of thing. If you're trying to hide, all of that calling out simply lets you know where the searchers are located, and you can avoid them fairly easily.
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(2) that if a child returns home they've more likely than not decided to go home before the designated cutoff of 5-7 minutes.
See above. You don't have to be deep in the woods to have a good hiding place.
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As an exaggerated example, if the group started praying and within 5 seconds the child opened the front door, it's almost impossible for the group prayer to have affected the child's decision to return home.
I think you're assuming a lot of things here. You can imagine a situation where the child is at the door, but afraid of what the parents will say/do now that the police are involved (or hears someone saying something upsetting in the house), and decides to turn around rather than confront the embarrassment/shame/whatever is associated with what has happened. I'm not going to say much on this end because there are all sorts of possibilities, as it's a very complex situation and I can't say with any level of confidence which class of scenarios is actually more likely.
The primary point that simplistic modeling is probably a pretty ineffective way of approaching the problem.