Abiraterone Acetate, Prednisone, and Apalutamide With or Without Ipilimumab or Cabazitaxel and Carboplatin in Treating Patients With Metastatic Castration-Resistant Prostate Cancer

Inclusion Criteria: Each patient must meet the following criteria to be enrolled in this study: 1. Willing and able to provide written informed consent. 2. Male aged 18 years and above. 3. Histologically or cytologically confirmed adenocarcinoma of the prostate. 4. Presence of metastatic disease documented on imaging studies (bone scan, CT and/or MRI scans). 5. Patients must have documented evidence of progressive disease as defined by any of the following: a) PSA progression: minimum of 2 rising values (3 measurements) obtained a minimum of 7 days apart with the last result being at least \>/= 1.0 ng/mL; b) New or increasing non-bone disease (RECIST 1.1 criteria); c) Positive bone scan with 2 or more new lesions (PCWG3). 6. Surgically or ongoing medically castrated, with baseline testosterone levels of ≤ 50 ng/dL (≤ 2.0 nM). 7. Eastern Cooperative Oncology Group (ECOG) Performance Status of ≤2. 8. Hemoglobin ≥ 7.5 g/dL in the presence of bone marrow involvement independent of transfusion and/or growth factors within 3 months prior to enrollment. 9. Platelet count ≥100,000/μL independent of transfusion and/or growth factors within 3 months prior to enrollment. 10. Serum albumin ≥3.0g/dL. 11. Medications known to lower the seizure threshold (see Appendix H under prohibited medications) must be discontinued or substituted at least 4 weeks prior to study entry. 12. Absolute neutrophil count (ANC) ≥ 1,500/mm3 13. Calculated creatinine clearance (Cockcroft-Gault Equation) ≥ 40 mL/min. 14. Serum potassium ≥ Institutional Lower Limit of Normal (ILLN). 15. Serum magnesium ≥ ILLN. 16. Serum bilirubin \< 1.5 x Institutional Upper Limit of Normal (IULN) (except for patients with known Gilbert's disease). 17. Serum AST or ALT \< 2.5 x IULN for patients without liver metastases. For patients with liver metastases AST or ALT \< 5 x IULN is allowed. 18. Able to swallow study drugs whole as a tablet/capsule. 19. Male subject with a female partner of childbearing potential or pregnant must agree to use two acceptable methods of contraception and not to donate sperm from time of Screening until 3 months after the last dose of study treatments as per Section 5.7. Patients must agree to tissue collection for correlative studies (including participation in PA13-0291 and PA13-0247 for MD Anderson participants) at the specified timepoints. (Additional details in Section 5.5.1.1). Exclusion Criteria: Patients who meet any of the following criteria will be excluded from the study: Prior or Concurrent Treatments: 1. Any prior treatment with: * Ipilimumab 2. Treatment within 28 days of Cycle1 Day1: * Any other systemic therapy for prostate cancer (with the exception of LHRH agonists and LHRH antagonists for testosterone suppression Sipuleucel-T, and bisphosphonates and RANK-ligand inhibitors for bone metastases which are allowed). * Any other investigational product. * Any medications listed in Appendix H 3. Treatment within 12 months of Cycle 1 Day 1 with any Cyp17-lyase inhibitor, any 2 nd generation AR antagonist (e.g. enzalutamide), cabazitaxel orcarboplatin. 4. Patients treated with any Cyp17-lyase inhibitor, any 2nd generation AR antagonist (e.g. enzalutamide), cabazitaxel or carboplatin whose disease progressed while on treatment or within 3 months of its discontinuation. (Patients who have received any of these treatments more than 12 months from study entry and whose disease did not progress while on treatment or within 3 months of its discontinuation are allowed on study). 5. Patients whose disease is refractory (defined as evidence of disease progression while on drug or within 3 months of its discontinuation) to more than 2 lines of chemotherapy. Any number of chemotherapies to which the patient's disease is not refractory are allowed, as long as time on treatment did not exceed 6 months (counted from day 1 of cycle 1 to day 1 of the last cycle of treatment). 6. Flutamide (Eulexin) treatment within 4 weeks of Cycle 1, Day 1 and Bicalutamide (Casodex) or nilutamide (Nilandron) within 6 weeks of Cycle 1 Day 1 Exceptions: if progression is documented prior to this time interval, or if patient is deemed by the treating physician to be highly unlikely to respond to AR inhibitor withdrawal (e.g. if PSA did not decline for ≥3 months in response to AR inhibitor given as a second line or later intervention, or if patient has symptoms attributable to disease progression) only a 3 day washout prior to Cycle 1, Day 1 will be required for any of them. 7. Radiation therapy for treatment of the primary tumor within 6 weeks of Cycle 1, Day 1. Patients who have received palliative radiation and recovered areeligible. 8. Any chronic medical condition requiring a higher dose of corticosteroid than 10mg prednisone/prednisolone daily. Use of inhaled, intranasal, intra-articular and topical steroids are acceptable, as is a short course (i.e. ≤ 1 day) of corticosteroids to prevent a reaction to the IV contrast used for CT scans. Concurrent illnesses: 9. Active infection (requiring oral or IV antibiotics or antiviral therapy) or other medical condition that would make prednisone/prednisolone (corticosteroid) use contraindicated. Known history of testing positive for human immunodeficiency virus (HIV) or known acquired immunodeficiency syndrome (AIDS). 10. A malignancy \[other than the one treated in this study\] which required radiotherapy or systemic treatment within the past 5 years, or has a ≥ 30% probability of recurrence within 24 months (except for non-melanoma skin cancer or Ta urothelial carcinomas). 11. Uncontrolled hypertension (systolic BP ≥ 140 mmHg or diastolic BP ≥ 90 mmHg). Patients with a history of hypertension are allowed provided blood pressure is controlled by anti-hypertensive treatment. 12. Prolonged QTc interval on pre-entry electrocardiogram (≥ 450 msec). 13. Known active or symptomatic viral hepatitis or chronic liver disease. 14. Clinically significant heart disease as evidenced by myocardial infarction, or arterial thrombotic events in the past 6 months, severe or unstable angina, history of clinically significant ventricular arrhythmias (such as ventricular tachycardia, ventricular fibrillation, or Torsade de pointes), New York Heart Association (NYHA) Class III-IV heart disease or cardiac ejection fraction measurement of \< 40% at baseline. 15. Autoimmune disease: Patients with a history of inflammatory bowel disease (including Crohn's disease and ulcerative colitis) and autoimmune disorders such as rheumatoid arthritis, autoimmune thyroiditis, autoimmune hepatitis, systemic progressive sclerosis \[scleroderma and variants\], Systemic Lupus Erythematosus, autoimmune vasculitis \[e.g., Wegener's Granulomatosis\] or autoimmune neuropathies (such as Guillain-Barre syndrome) are excluded from this study. Vitiligo and adequately controlled endocrine deficiencies such as hypothyroidism are not exclusionary. 16. Patients who have had a history of illness which put them at current risk for bowel perforation such as acute diverticulitis, intra-abdominal abscess, GI obstruction and abdominal carcinomatosis. 17. History of seizure or known condition that may pre-dispose to seizure (including but not limited to prior stroke or loss of consciousness within 1 year prior to randomization, brain arteriovenous malformation; or intracranial masses such as schwannomas and meningiomas that are causing edema or mass effect). 18. Gastrointestinal disorder affecting absorption. Any underlying medical or psychiatric condition, which in the opinion of the Investigator, will make the administration of study drug hazardous or obscure the interpretation of adverse events. 19. Untreated symptomatic spinal cord compressions. Other: 20. Prisoners or subjects who are involuntarily incarcerated. 21. Subjects who are compulsorily detained for treatment of either a psychiatric or physical (e.g. infectious disease) illness.