A context-free grammar for all strings that end in b and have an even number of bs

I'm trying to find CFG's that generate a regular language over the alphabet {a b}

I believe I got this one right: All strings that end in b and have an even number of b's in total:

$\qquad S \to SS \\
\qquad S \to YbYb \mid \varepsilon \\
\qquad Y \to aY \mid \varepsilon$

However, Im not sure how to accomplish this with an odd number of b's.

So for example, how could I find a CFG that generates all strings that end in b and have an odd number of b's in total: So far I have this,

$\qquad S \to SS \\
\qquad S \to YYb \mid \varepsilon \\
\qquad Y \to abY \mid baY \mid \varepsilon$

But this can generate abababb so it's incorrect and Im stumped at this point.

There is an easy and very natural way how to derive a context-free grammar $G=(V,\Sigma,R,S)$ from a DEA $M=(Q,\Sigma,\delta,q_0,F)$. We set

• $V=Q$

• $S= q_0$

• $R$ consists of the following rules:

• for all $X\in Q$ and $a\in \Sigma$ with $Y=\delta(X,a)$ we add the rule $X \to aY$

• for all $X\in F$ we add the rule $X \to \varepsilon$

• there are no other rules in $R$

It's an easy exercise to show that $L(G)=L(M)$. With this technique you can now build your automaton first (which is an easy two-state example in your case), and then apply this construction.

StackExchange Computer Science Q#6163, answer score: 4