Carboplatin and Paclitaxel With or Without Bevacizumab Compared to Docetaxel, Carboplatin, and Paclitaxel in Treating Patients With Stage II, Stage III, or Stage IV Ovarian Epithelial, Fallopian Tube, or Primary Peritoneal Cavity Carcinoma (Cancer)

Inclusion Criteria: * Histologically confirmed fallopian tube, ovarian epithelial, or primary peritoneal carcinoma * Stage II-IV disease * The following epithelial cell types are allowed: * Carcinosarcoma * Serous adenocarcinoma * Endometrioid adenocarcinoma * Mucinous adenocarcinoma * Undifferentiated carcinoma * Clear cell adenocarcinoma * Mixed epithelial carcinoma * Transitional cell carcinoma * Malignant Brenner's tumor * Adenocarcinoma not otherwise specified * Must have undergone prior surgery for ovarian or peritoneal carcinoma within the past 12 weeks * Optimal (≤ 1 cm residual disease) or suboptimal residual disease following initial surgery * Must have a procedure for determining diagnosis of ovarian/peritoneal carcinoma with appropriate tissue for histologic evaluation * Synchronous primary endometrial cancer or prior endometrial cancer allowed provided the following criteria are met: * Stage ≤ IB * Less than 3 mm invasion without vascular or lymphatic invasion * No poorly differentiated subtypes (e.g., grade 3, clear cell, or papillary serous) * No epithelial ovarian carcinoma of low malignant potential (borderline carcinomas) * No CNS disease (e.g., seizures not controlled with standard medical therapy) or metastasis * GOG performance status 0-2 * Absolute neutrophil count ≥ 1,500/mm\^3 * Platelet count ≥ 100,000/mm\^3 * INR ≤ 1.5 (or an in-range INR, usually between 2 and 3, if a patient is on a stable …