Regular expression for strings with an even number of zeroes and ones

Question

What is a regular expression for strings of 0 and 1 with an even number of zeros and an even number of ones?

I have something like (1*01*01*)*(0*10*10*)*

Answer 1

1100 is in the language, but doesn't match your expression. 10101 is not in the language, but your expression matches it.

I'd suggest starting by drawing a DFA. There's a pretty obvious 4-state machine that recognizes this language. (Is it possible to do better?) The empty string is in the language, so the start state is an accepting state. Are there other accepting states? For a non-accepting state S, is there a prefix that takes you from start->S? Is there a way to loop from S back to S without hitting an accepting state? Is there suffix that takes you from S back to an accepting state?