I am trying to write a regular expression for the set of all strings in $\Sigma^*$ that starts with an even number of $b$'s and contains at most two $a$'s. The language contains only $a$'s and $b$'s.
This is what I have figured out so far:
$b^*$ - language with exactly $0$ $a$'s;
$b^*ba^*$ - language with exactly one $a$;
$b^*ba^*ba$ - language with exactly two $a$'s;
$bb^*$ - language starting with an even number of $b$'s.
I'm not sure where to go with this (or if how I have defined the language starting with an even number of $b$'s right). Can anyone help point me in the right direction?