The purpose of this study is to investigate the efficacy, safety and tolerability of eltrombopag in children with previously treated chronic immune thrombocytopenia who are between 1 and 17 years of age. This is a 2 part study. In part 1, patients will be randomized to receive either eltrombopag or placebo for 13 weeks. All patients who complete part 1 will enter part 2. In part 2, all patients will receive 24 weeks of eltrombopag.