Will my 4 yr old daughter likely be allergic to bee stings if her mother has anaphylaxis to bee stings? She gets a rash with amoxil (amoxicillin).

Bee sting. No your daughter is not more likely to have a severe allergic reaction to bee stings. There is rash associated with Amoxicillin called the Amoxicillin rash. It is a rough or bumpy rash sometimes described as feeling like sand paper. It is not an allergic reaction to amoxicillin. If daughter rash to Amoxicillin was of this type she is not likely allergic to amoxicillin.
Not likely. There is a very weak relationship between a family member having bee sting allergy & increased risk for other family members. If your daughter has not had an anaphylactic reaction to bee stings she needs neither skin nor blood testing nor an Epipen (epinephrine) junior. She should be tested only if she has had hives, difficulty breathing, or passing out after a bee, hornet or wasp sting.
Uncertain. There is no guarantee that she will develop the same allergy as her mother, and the rash with Amoxicillin also does not suggest what, if anything else, she is allergic to. The best advice would be to see your physician who can perform either rast testing (blood test) or skin prick testing while keeping the mothers history in mind.