Can I use my home insurance for a slip and fall injury in my back door from the snow?

No, because you can't make a liability claim against yourself. You'll need health insurance in order to cover the injury.

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Here's another example: Say you own two cars, and you accidentally back one into the other in the driveway. You can't use your liability coverage for the damage because you'd be making a liability claim against yourself. (You would need collision coverage for the damage you do to your own car.)