She still has a much better chance than if she was unarmed. Especially since it's here home turf and she'd have a defensive advantage.
If the burglar has good reason to assume she's armed, they'll go to the next house.
That's simply not how it works. When people are going to rob someone, they'll typically bring a firearm unless they're meth heads without a plan. I speak from experience; having a gun doesn't do anything in a home invasion unless you happen to defend your house military-style at all times [or are lucky enough to be in a position to respond immediately to a threat]. I got out of my shower to my friends being held hostage at gunpoint in my basement, waiting for me to get out of the shower so they could rob us all at once. I own two rifles and I'm trained at self defence. I consider myself more than prepared; but often you do not want to give the criminals
a reason to kill you. And I doubt your grandmother would handle such a situation as well as I did, without getting someone shot.
Telling people to fight fire with fire in such cases is argueably
more dangerous, since you'd be losing your life or getting injured instead of simply losing
stuff.
That's why the arguement is a good example of an Occam without a real answer. Simple may not be the best solution in the case of gun regulation. I advocate more people arming themselves, but it's a double edged sword in the case of violent robberies. Hence why the issue is such a tangled clusterfuck of opinion.