The purpose of this study is to compare the effects of a drug called apixaban to aspirin in patients with atrial fibrillation (irregular heart rate) and a recent brain hemorrhage to determine which is more effective for the prevention of future strokes. Participants will be randomly assigned to 1 of 2 groups. Group 1 will receive aspirin 81 milligrams once per day, plus a placebo (inactive) pill twice per day. Group 2 will receive apixaban 5 milligrams twice per day, plus a placebo pill once per day. Apixaban is approved by the U.S. Food and Drug Administration (FDA) for stroke prevention in patients with atrial fibrillation, but its use in this study is experimental.