Packages are insured so, unless they suspect fraud, they will let insurance cover it.

Consumers usually pay FedEx for insurance when we need a package but I bet Amazon self insures, given their size. Either way, money is set aside to cover situations like this.

I know insurance covers it; I'm wondering how they determine what situations are likely to be fraudulent. Is customer history involved? My parents have never ordered from Amazon; if the same thing happens to them on their first purchase, what would the resolution be?

They probably give users the benefit of the doubt the first time, then flag them if they do it too much. At the scale of amazon, the cost of a product is less then the cost of the time to chase up every single report of theft.