This study is testing a new medicine called efgartigimod to see if it is safe and works well for people with a blood condition called primary immune thrombocytopenia, or ITP. These are people who tried other treatments but did not get better. In the study, people will be randomly placed into one of two groups. One group will get the real medicine, and the other group will get a placebo, which is a fake medicine. Both are given through a tube in the arm called an IV. For the first 4 weeks, everyone gets either the medicine or the placebo. After that, depending on their blood test results, they will keep getting it every week or every other week for 20 more weeks. After 24 weeks, people can join the next part of the study where everyone gets the real medicine for up to 52 weeks. If someone does not get better after 12 weeks, they may be able to switch to the real medicine sooner.