Clopidogrel (Plavix) and aspirin are two antithrombotic agents (blood thinners) commonly used in patients with previous thrombotic events (stroke or heart attack). Thrombosis is the formation of a blood clot in a blood vessel. Patients with polycythemia vera are routinely treated with aspirin which has been shown to be effective in reducing their thrombotic risk. However, in polycythemia vera patients with previous thrombosis, a further benefit might be obtained by using the combination of aspirin and clopidogrel which is routinely used in patients with recent acute myocardial ischemia (reduced blood supply to the heart muscle). The study will assess whether this combination therapy greatly increases the risk of bleeding versus aspirin alone, if clopidogrel reduces biological factors that might lead to a stroke or heart attack, and whether a high number of patients with polycythemia vera are resistant to clopidogrel.