Carboplatin and Paclitaxel With or Without Bevacizumab in Treating Patients With Newly Diagnosed Ovarian Epithelial Cancer, Fallopian Tube Cancer, or Primary Peritoneal Cavity Cancer

DISEASE CHARACTERISTICS: * Histologically confirmed ovarian epithelial, fallopian tube, or primary peritoneal cavity cancer * Newly diagnosed disease * Meets 1 of the following staging criteria: * High-risk stage I or IIA disease (grade 3 disease or clear cell carcinoma only) * Stage IIB-IV disease (all grades and all histological types) * Must have undergone initial surgery (e.g., debulking cytoreductive surgery or a biopsy if the patient has stage IV disease) within the past 6 weeks * Patients with stage IV disease for which initial surgical debulking was not appropriate are eligible provided the following criteria are met: * Stage IV disease diagnosed by histology * No planned surgery prior to disease progression, including interval debulking surgery * Patients with prior early-stage ovarian epithelial or fallopian tube carcinoma treated with surgery alone are eligible at the time of diagnosis of abdominopelvic recurrence provided no further interval cytoreductive therapy is planned prior to disease progression * Synchronous primary endometrial carcinoma or a past history of primary endometrial carcinoma allowed provided the following criteria are met: * Disease ≤ stage IB * No more than superficial myometrial invasion * No lymphovascular invasion * Not poorly differentiated (i.e., no grade 3, papillary serous, or clear cell disease) * Measurable or nonmeasurable disease * No ovarian nonepithelial cancer, including malignant mixed Müllerian tumors * No borderline tumors (e.g., tumors of low malignant potential) * No history or clinical suspicion of brain metastases or spinal cord compression * CT scan or MRI of the brain is mandatory (within 4 weeks prior to randomization) in case of suspected brain metastases * Spinal MRI is mandatory (within 4 weeks prior to randomization) in case of suspected spinal cord compression PATIENT CHARACTERISTICS: * ECOG performance status 0-2 * Life expectancy \> 12 weeks * ANC ≥ 1,500/mm\^3 * Platelet count ≥ 100,000/mm\^3 * Hemoglobin ≥ 9 g/dL (can be post-transfusion) * INR ≤ 1.5 * APTT ≤ 1.5 times upper limit of normal (ULN) * Bilirubin ≤ 1.5 times ULN * ALT and AST ≤ 2.5 times ULN * Creatinine ≤ 2.0 mg/dL * Proteinuria ≤ 1+ by urine dipstick OR ≤ 1 g by 24-hour urine collection * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception during and for ≥ 6 weeks after completion of study therapy * No significant traumatic injury within the past 4 weeks * No cerebrovascular accident, transient ischemic attack, or subarachnoid hemorrhage within the past 6 months * No other malignancies within the past 5 years except for adequately treated carcinoma in situ of the cervix, and/or basal cell skin cancer, and/or early endometrial carcinoma * No pre-existing sensory or motor neuropathy ≥ grade 2 * No history or evidence of CNS disease (e.g., uncontrolled seizures) by neurological examination unless adequately treated with standard medical therapy * No history or evidence of thrombotic or hemorrhagic disorders * No uncontrolled hypertension (i.e., blood pressure \> 150/100 mm Hg despite antihypertensive therapy) * No known hypersensitivity to bevacizumab and its excipients, chemotherapy, or Cremophor EL * No nonhealing wound, ulcer, or bone fracture * Patients with granulating incisions healing by secondary intention with no evidence of facial dehiscence or infection are eligible but require three weekly wound examinations * No clinically significant cardiovascular disease, including any of the following: * Myocardial infarction or unstable angina within the past 6 months * New York Heart Association class II-IV congestive heart failure * Poorly controlled cardiac arrhythmia despite medication * Rate-controlled atrial fibrillation allowed * Peripheral vascular disease ≥ grade 3 (i.e., symptomatic and interfering with activities of daily living requiring repair or revision) * No evidence of other disease or condition, metabolic dysfunction, physical examination findings, or laboratory findings that would contraindicate the use of an investigational drug or put the patient at high-risk for treatment-related complications PRIOR CONCURRENT THERAPY: * See Disease Characteristics * More than 4 weeks since other prior surgery or open biopsy * No prior systemic therapy for ovarian cancer (e.g., chemotherapy, monoclonal antibody therapy, tyrosine kinase inhibitor therapy, or hormonal therapy) * Prior adjuvant chemotherapy allowed for other malignancies (e.g., breast or colorectal carcinoma) if malignancy was diagnosed over 5 years ago with no evidence of subsequent recurrence * No prior mouse CA 125 antibody * No prior radiotherapy to the abdomen or pelvis * More than 10 days since prior and no concurrent chronic use of acetylsalicylic acid (\> 325 mg/day) * Low-dose (\< 325 mg/day) acetylsalicylic acid allowed * More than 10 days since prior and no concurrent full-dose oral or parenteral anticoagulants or thrombolytic agents for therapeutic purposes * Use of therapy for line patency allowed provided INR \< 1.5 * More than 30 days since prior and no other concurrent investigational agent or participation in another clinical trial * No other concurrent systemic antitumor agents * No concurrent surgery * No concurrent maintenance chemotherapy or intraperitoneal chemotherapy (including cytotoxic chemotherapy)