Study of Pembrolizumab and Chemotherapy With or Without Radiation in Small Cell Lung Cancer (SCLC)

Inclusion Criteria: * Participants must have histologically confirmed small cell lung carcinoma not amenable to initial concurrent radiotherapy (extensive-stage disease). * Participants may have evaluable or measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as ≥20 mm with conventional techniques or as ≥10 mm with spiral CT scan, MRI, or calipers by clinical exam. * Participants in cohort B must have completed 1 cycle of systemic chemotherapy. Therapy with the combination must start no sooner than 3 weeks from the last dose of chemotherapy and no later than 5 weeks from the last dose of chemotherapy. Participants in cohort B must not have had progression of disease prior to the start of therapy. * Participants in cohort C must have completed systemic therapy (4-6 cycles cisplatin or carboplatin + etoposides) and NOT be a candidate for consolidation thoracic radiotherapy or PCI. Participants in cohort C must initiate therapy with pembrolizumab within 6 weeks of the last dose of chemotherapy (therapy must not start within 2 weeks from the last dose). Participants in cohort C must not have had progression of disease prior to the start of therapy. * Participants in cohort D must have completed systemic therapy AND have completed either consolidation thoracic radiotherapy or PCI or both completed either consolidation thoracic radiotherapy or PCI or both. Participants in cohort D must initiate therapy with pembrolizumab within 6 weeks of the last dose of radiation. Therapy must not start within 2 weeks from the last dose. Consolidation radiotherapy dose must NOT be more than 3000 centigray (cGy). Participants in cohort D must not have had progression of disease prior to the start of therapy. * Age \> 18 years. * Eastern Cooperative Oncology Group (ECOG) performance status ≤2 (Karnofsky ≥60%) * Life expectancy of greater than 3 months * Participants must have normal organ and marrow function during screening and on Cycle 1, day 1 as defined below. \* Adequate Organ Function Laboratory Values * System Laboratory Value Hematological * Absolute neutrophil count (ANC) ≥1,500 /microliter (mcL) * Platelets ≥100,000 / mcL * Hemoglobin ≥9 g/dL or ≥5.6 mmol/L without transfusion or erythropoietin (EPO) dependency (within 7 days of assessment) Renal * Serum creatinine OR Measured or ≤1.5 X upper limit of normal (ULN) OR calculated creatinine ≥60 mL/min for subject with creatinine clearance (GFR can also be used levels \>1.5X institutional ULN in place of creatinine or CrCl) Hepatic * Serum total bilirubin ≤ 1.5 X ULN OR Direct bilirubin ≤ ULN for subjects with total bilirubin levels \> 1.5 ULN * AST (SGOT) and Alanine transaminase (ALT) (SGPT) ≤ 2.5 X ULN OR ≤ 5 X ULN for subjects with liver metastases * Albumin \>2.5 mg/dL Coagulation * International Normalized Ratio ≤1.5 X ULN unless subject is receiving (INR) or Prothrombin Time (PT) anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants * Activated Partial Thromboplastin ≤1.5 X ULN unless subject is receiving Time (aPTT) anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants. * Availability of a diagnostic or pre-chemotherapy tissue biopsy is required (cytologic specimens or bone biopsies not accepted). This biopsy must be within 6 weeks of starting initial therapy. A minimum of 205 μm slides or block is required. * Participants in cohorts B-D must be willing to provide tissue from a newly obtained core or excisional biopsy of a tumor lesion. Newly-obtained is defined as a specimen within 4 weeks to initiation of treatment and AFTER the last dose of any prior therapy. * Participants with treated brain metastases are allowed. Radiation must be completed at least 2 weeks prior to pembrolizumab dosing and participants must not require ongoing steroids. Participants with untreated brain metastases that are all \<5 mm with no clinical symptoms or vasogenic edema may be allowed on study on a case-by-case basis on discussion with sponsor. These participants will require MRI monitoring every 6 weeks to ensure stability. * The effects of pembrolizumab on the developing human fetus are unknown. For this reason and because the chemotherapy and radiation also used in this trial are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) or be surgically sterile prior to study entry and for the duration of study participation. Subjects of childbearing potential are those who have not been surgically sterilized or have not been free from menses for \> 1 year. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. * Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of chemotherapy, radiation, and pembrolizumab administration. * Ability to understand and the willingness to sign a written informed consent document. Exclusion Criteria: * Participants in cohort A may not have had prior therapy for their disease. Participants in cohort B may not have had more than 1 cycle of systemic therapy (cisplatin or carboplatin + etoposide). Participants in cohort C and D should not have had more than one prior regimen of chemotherapy. * For participants entering cohorts C or D, prior treatment-related toxicities should have resolved to grade 1 or baseline (with the exception of anemia (as per inclusion criteria, alopecia, and neuropathy (\< grade 2 allowed). * Participants who have had a CR after pre-study therapy are not eligible for study. * No thoracic radiation \> 3000 cGy allowed. * Prior radiation or surgery must have completed at least 2 weeks prior to initiation of therapy and all toxicities or complications from these must have resolved to baseline or grade 1 prior to starting therapy (with the exception of anemia (as per inclusion criteria, alopecia, and neuropathy (\< grade 2 allowed). * No stroke, myocardial infarction, or major surgery within 3 months of starting on therapy * Participants who are receiving any other investigational agents or have received investigational therapy or any anti-cancer monoclonal antibody (mAB) within 4 weeks prior to the 1st dose of pembrolizumab. * Has received prior therapy with an anti-PD-1, anti-PD-L1, or anti-PD-L2 agent * Has active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (eg., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. * Has known history of non-infectious pneumonitis which required steroids, or any evidence of current, non-infectious pneumonitis. * Has an active infection requiring systemic therapy. * Has a diagnosis of immunodeficiency or is receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment. * Has a known history of active Bacillus Tuberculosis (TB) * History of allergic reactions attributed to compounds of similar chemical or biologic composition to pembrolizumab, cisplatin, carboplatin, or etoposide. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. * Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial. * Is pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the trial, starting with the pre-screening or screening visit through 120 days after the last dose of trial treatment. * Has known active Hepatitis B (e.g., HBsAg reactive) or Hepatitis C Virus (HCV) (e.g., HCV RNA \[qualitative\] is detected). * Has a known history of Human Immunodeficiency Virus (HIV) (HIV 1/2 antibodies). * Has received a live vaccine within 30 days of planned start of study therapy Note: Seasonal influenza vaccines for injection are generally inactivated flu vaccines and are allowed; however intranasal influenza vaccines (e.g., Flu-Mist are live attenuated vaccines, and are not allowed. * Has a known additional malignancy that is progressing or requires active treatment or has required active treatment within the last 2 years. Exceptions include basal cell carcinoma of the skin or squamous cell carcinoma of the skin that has undergone potentially curative therapy or in situ cervical cancer or in situ bladder cancer. * Has a paraneoplastic syndrome other than SIADH (hyponatremia). * Evidence of interstitial lung disease.