[![FA][3]][3]  
$r_1=\Lambda+(a+b)^*b$   
$r_2=(b+aa^*b)^*$   
$r_3=b+\$+aa^*b+(b+\$+aa^*b)(b+aa^*b)*(b+\$+aa^*b)$  

For this FA, which i think of as accepting "$\Lambda$ or anything ending in $b$", i came up with $r_1$, then i used [FSM2Regex][1] which gave me $r_3$ which i simplified into $r_2$ using [regex simplifier][2].  
now $r_2$ seems better than $r_1$. **should i go with it? if so, what is lacking in $r_1$? i can't see the difference b/w $(a+b)^*b$ from $r_1$ AND $(aa^*b)^*$ from $r_2$. is one the subset of other? are they overlapping or disjoint?**  
Inputting any of the three RE's back into [FSM2Regex][1] gives RIDICULOUSLY complex TG's esp cuz for reason that is beyond me, it also draws `$` (null string) transitions. even $r_3$ doesn't give back the FA i used to generate it. **so question is : are $r_1$ and $r_2$ different? how? what should i go with?**    


  [1]: http://ivanzuzak.info/noam/webapps/fsm2regex/
  [2]: http://ivanzuzak.info/noam/webapps/regex_simplifier/
  [3]: https://i.sstatic.net/p0COX.png