Suppose there is a set S of 1000 elements. Given sets A, B, C, D are subset of S and of cardinality 200, 300, 400, 500 respectively. Suppose further that A,B,C,D are made by drawing elements uniformly and independently from S. What is the expected size of $A\cup B\cup C\cup D$?
I calculated it as follow: take the 200 elements from A, then it covers 20% of S. We can expect 80% of B are not covered. So we have (200+0.8*300)=440 from $A\cup B$. Arguing this way, I got the expected size of $A\cup B\cup C\cup D$ is 832.
Am I right? Can anyone give a formal proof for this?