This paper provides a simple hybrid design and numerical analysis of the graphene-coated fiber-optic surface plasmon resonance (SPR) biosensor for breast cancer gene-1 early onset (BRCA1) and breast cancer gene-2 early onset (BRCA2) genetic breast cancer detection. Two specific mutations named 916delTT and 6174delT in the BRCA1 and BRCA2 are selected for numerical detection of breast cancer. This sensor is based on the technique of the attenuated total reflection (ATR) method to detect deoxyribonucleic acid (DNA) hybridization along with individual point mutations in BRCA1 and BRCA2 genes. We have numerically shown that momentous changes present in the SPR angle (minimum: 135 % more) and surface resonance frequency (SRF) (minimum: 136 % more) for probe DNA with various concentrations of target DNA corresponding to a mutation of the BRCA1 and BRCA2 genes. The variation of the SPR angle and SRF for mismatched DNA strands is quite negligible, whereas that for complementary DNA strands is considerable, which is essential for proper detection of genetic biomarkers (916delTT and 6174delT) for early breast cancer. At last, the effect of electric field distribution in inserting graphene layer is analyzed incorporating the finite difference time domain (FDTD) technique by using Lumerical FDTD solution commercial software. To the best of our knowledge, this is the first demonstration of such a highly efficient biosensor for detecting BRCA1 and BRCA2 breast cancer. Therefore, the proposed biosensor opens a new window toward the detection of breast cancers.