Arizona is a strict liability state for dog bites
The owner is liable for a bite regardless of whether the dog had ever shown aggression and regardless of whether the owner knew of any risk. There is no first-bite allowance. That removes most of the defenses owners assume they have and makes the size of the claim the only real variable.
Carriers restrict this exposure in writing
Some policies exclude certain breeds, some exclude any dog with a bite history, and some cap the liability for animals at a figure well below the policy limit. These restrictions sit in the form and are rarely mentioned at the quote, which is why they surprise people.
The umbrella has to agree too
An umbrella generally will not cover an exposure the underlying policy excludes. If the homeowners form excludes the dog, the umbrella above it does not fill that gap. Both layers need checking rather than just the top one.