A Study Comparing Personalized Radiation Therapy With Standard Radiation Therapy in People With HPV-Positive Throat Cancer

Inclusion Criteria: * Pathologically (histologically or cytologically) proven diagnosis of HPV associated squamous cell carcinoma of the oropharynx (tonsil, base of tongue, or oropharyngeal walls) or squamous cell carcinoma with an unknown primary. Surgical removal of primary site is allowed. * Patients must test positive for both p16 expression (70% nuclear and cytoplasm expression; Ventana Medical Systems) and mRNA HPV in situ hybridization (RNAscope® 2.5 HD Reagent kit (Advanced Cell Diagnostics, Inc, Hayward, CA). Any CLIA certified testing method can be used. * Clinical stage Tx-2, N1-2c (AJCC, 7th ed.) without evidence of distant metastasis based on FDG PET/CT. * ECOG Performance Status of 0-1 or KPS \>/=70 * Age ≥ 18 * Adequate hematologic function within 30 days prior to registration, defined as follows: * White Blood Count (WBC) ≥ 2 K/mcL * Absolute neutrophil count (ANC) ≥ 1,500 cells/mm\^3 * Platelets ≥ 100,000 cells/mm\^3 * Hemoglobin ≥ 10.0 g/dl * Adequate renal function within 30 days prior to registration, defined as follows: o Serum creatinine \< 1.5 mg/dl or creatinine clearance (CC) ≥ 50 ml/min determined by 24-hour collection or estimated by Cockcroft-Gault formula: CCr male = \[(140 - age) x (wt in kg)\] \[(Serum Cr mg/dl) x (72)\] CCr female = 0.85 x (CrCl male) * Negative serum pregnancy test within 14 days prior to registration for women of childbearing potential * The patient must provide study-specific informed consent prior to study entry * Optional section of the protocol: Patients must be able to undergo MRI scans, i.e. not claustrophobic Exclusion Criteria: * Patients with prior head and neck radiation therapy where there is \>30% overlap with the current head and neck radiation fields. Exceptions can be made if determined by the PI/Co-PI that the patient can proceed with protocol activities * Patients whose tumors are borderline T4 based on anterior tumor extension to the extrinsic muscles of the tongue * Patients with simultaneous primary cancers outside of the oropharynx o Note: Exceptions can be made for patients with simultaneous primaries outside the oropharynx if determined by the PI/Co-PI the patient can proceed with protocol activities * Prior invasive malignancy (except non-melanomatous skin cancer) unless disease free for 3 years or if cure rate from treatment at 5 years to be 90% or greater o Note: Exceptions can be made for patients with prior malignancies outside the oropharynx if determined by the PI/Co-PI the patient can proceed with protocol activities. * Prior systemic chemotherapy for the study cancer; note that prior chemotherapy for a different cancer is allowable * No particle therapy * Patients who are deemed non-compliant to all the protocol related activities * Contraindications to receive either cisplatin or the combination of carboplatin/5-fluorouracil at the prescribed doses. * Severe, active co-morbidity defined as follows: * Unstable angina and/or congestive heart failure requiring hospitalization within the last 6 months * Transmural myocardial infarction within the last 6 months * Acute bacterial or fungal infection requiring intravenous antibiotics at the time of registration * Chronic Obstructive Pulmonary Disease exacerbation or other respiratory illness requiring hospitalization or precluding study therapy within 30 days of registration * Hepatic Insufficiency resulting in clinical jaundice and/or coagulation defects