Language of all strings that has exactly 1 triple b

Question

I am new to automata and learning to make regular expression for languages. But I have been stuck on this one.

Suppose we have a language L, Language of all strings that has exactly 1 triple “b” defined over alphabet set Σ = {a, b}
Now after several tries, I came up with this
(a* (ab)* (ba)* )* bbb (a* (ab)* (ba)* )* but then I realize that this is wrong too because the string abbbabababb doesn't fit on this.

Kindly someone point out at my mistake or help me solve it as I have spent almost an hour on this.

Answer 1

Start with a DFA that describes the language:

States: start, b, bb, bbb, bbba, bbbab, bbbabb, x
Accepting: bbb, bbba, bbbab, bbbabb
Transitions: 
  start, a, start
  start, b, b
  b, a, start
  b, b, bb
  bb, a, start
  bb, b, bbb
  bbb, a, bbba
  bbb, b, x
  bbba, a, bbba
  bbba, b, bbbab
  bbbab, a, bbba
  bbbab, b, bbbabb
  bbbabb, a, bbba
  bbbabb, b, x

(Example of this DFA in action)

From there, you can convert the DFA into a regular expression, though you'll omit the x 'dump' state when using this algorithm. You'll end up with an expression something like

(|a|ba|bba)*(bbb|bbba(|a|ba|bba)*(|b|bb))