Am I correct? (Finite Automata)

I was given a regular expression, and I am suppose to covert it to NFA and then DFA. Here's the regular expression:

a ( b | c )* a | a a c* b

Then I coverted this to NFA using Thomson's algorithm:

and here's the DFA:

Can someone please take a quick look at let me know if I am wrong or right?

Since this is very likely homework, I'm hesitant to just give you the complete correct solution.

Your NFA appears correct, but has a lot of superfluous states that aren't necessary but do not adversely affect its correctness. (At first glance it looks like you could remove 11 states.)

Your DFA is incorrect, though. This is because when you branch off to begin handling one condition of the string or the other, you later rejoin them together. This allows it to take the path from an accepted string matching a(b|c)*a and take in another b or c by travelling to nodes 15,17 or 11. It then accepts this string even though it doesn't match your expression.

What you need to do is basically stop this from happening. If you have additional questions feel free to ask.

I highly recommend making a list of test strings that you know should be and shouldn't be accepted, and then trace them through, making sure your automata ends in the correct (accept or reject) state.