MEDI4736 With Selumetinib for KRAS Mutant Non-Small Cell Lung Cancer (NSCLC)

Study Groups: If participant is found to be eligible to take part in this study, participant will be randomly assigned (as in the flip of a coin) to 1 of 2 study groups. This is done because no one knows if one study group is better, the same, or worse than the other group. * If participant is in Group 1, participant will take selumetinib 2 times each day. * If participant is in Group 2, participant will take selumetinib 2 times each day on Days 1-7 and 15-22 of each study cycle. All participants will also receive MEDI4736. Study Drug Administration: Each study cycle is 28 days. Participant should take each dose of selumetinib about 12 hours apart after fasting (having nothing to eat or drink except water) for 1 hour before and 2 hours after taking selumetinib. Participant will follow the dosing schedule to which participant has been assigned (as described above under Study Groups). Participant should record each selumetinib dose in the dosing diary that will be given to participant. Participant should bring the diary to participant's study visits at the end of every cycle. Both groups will receive MEDI4736 by vein over about 60 minutes on Day 1 of every cycle. Participant will be given standard drugs to help decrease the risk of side effects. Participant may ask the study staff for information about how the drugs are given and their risks. Length of Study: Participant may receive the study drugs for as long as the doctor thinks is in participant's best interest. Participant will no longer be able to take the study drugs if the disease gets worse, if intolerable side effects occur, or if participant is unable to follow study directions. Participation on the study will be over after the Follow-Up Period. Study Visits: About 14 days before the first dose of the study drugs: * Participant will have a physical exam. * Blood (about 3 teaspoons) and urine will be collected for routine tests. * If the doctor thinks it is needed, participant will have an EKG. On Day 1 of every cycle: * Participant will have a physical exam. * Blood (about 3 teaspoons) and urine will be collected for routine tests. * If the doctor thinks it is needed, participant will have an eye exam. * If participant can become pregnant and the doctor thinks it is needed, blood (about ½ teaspoon) or urine will be collected for a pregnancy test. On Day 1 of Cycle 1 and at Week 16, participant will have an EKG. Participant will have triplicate EKGs on Day 1 of Cycle 1. At Week 8 and about every 8 weeks after that, participant will have a CT scan or MRI to check the status of the disease. At Week 12 and about every 12 weeks after that, participant will have an ECHO or multigated acquisition (MUGA) scan. End-of-Study Visits: At about 30 days and 90 days after participant's last dose of the study drugs: * Participant will have a physical exam. * Blood (about 3 teaspoons) will be drawn for routine tests. * Participant will have an EKG. * If participant can become pregnant and the doctor thinks it is needed, blood (about ½ teaspoon) or urine will be collected for a pregnancy test. Follow Up Period: About every 6 months, participant will be called by a member of the study staff to ask how participant is doing and if participant has started any new treatments outside of the study. These calls should last about 10 minutes each time. If the doctor thinks it is needed, participant may return to the clinic for either a CT scan or MRI to check the status of the disease. This is an investigational study. MEDI4736 and selumetinib are not FDA approved or commercially available. They are currently being used for research purposes only. The study doctor can explain how the study drugs are designed to work. Up to 76 participants will be enrolled in this study. All will take part at MD Anderson.