Having a family history, your risk of cancer is slightly increased, but it doesn't mean that you'll also develop this disease, as the majority of women who develop breast cancer have no known risk factors at all.
You don't state what your Mammogram test points out, but what has been found seems suspicious and only a biopsy will give a definite answer.
BIRADS is a quality control system, used by radiologists after interpreting Mammograms, Ultrasounds and MRIs. This system allows radiologist to categorize the findings by numbers. The scores go from 0 to 5.The higher the number the more suspicious is the abnormality.
Now, BI-RADS Category 4 means that there is a possibility that something doubtful is going on but nothing more than that. Your score is low since it’s 4 A. However a biopsy is usually recommended to check the suspicious area. I would also like to let you that many biopsies performed on suspicious findings with BIRADS score 4, turn out benign and I hope the same for you too.
Best wishes and good luck!