On the right path? Finite automaton diagram for a regular language

Question

Hello, I am trying to show that a language L is regular through a regular expression and a finite automaton.

The expression makes sense, but wanted to see if I am on the right path for the finite automata. I feel it could be simplify and there doesn't need to be so many states.

Regular Expression

We will describe language L:

1. start with all the binary words that have all the zeroes before all the ones
2. add all the words that have exactly two ones
3. then add all the words that have an even number of zeros

To prove its regular, we can use the sub-info above to form a combined expression.

• R1: 0∗1∗
• R2: 0∗10∗10∗
• R3: (1∗01∗01∗)∗

RG = (0∗1∗) + (0∗10∗10∗) + ((1∗01∗01∗)∗)

Finite Automata