This is a two part, double-blind, randomized, placebo-controlled and open-label Phase III study to investigate the efficacy, safety and tolerability of eltrombopag in pediatric patients with previously treated chronic ITP. In Part 1, patients will be randomized to receive eltrombopag or placebo in a 13-week double-blind, placebo-controlled treatment period. After completing Part 1, patients will begin Part 2, in which they will receive eltrombopag in an open-label manner during a 24-week treatment period.