Krdytkyu

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.

To be clearer, we use "triple $b$'s" to mean three consecutive $b$'s.

What you would like to figure out first is a detailed description or characterization of a string that has not triple $b$'s and that does not end at an $b$, the part of the string that is before that triple $b$.

That string should start with zero or more $a$'s, possibly followed by one or two $b$'s followed by one or more $a$'s, possibly followed by one or two $b$'s followed by one or more $a$'s, and so on for some rounds. That is, $a^*((bmid bb)aa^*)^*$.

So, a regular expression could be $a^*((bmid bb)aa^*)^*bbb(aa^*(bmid bb))^*a^*$, or written symmetrically, $a^*((bmid bb)aa^*)^*bbb(a^*a(bbmid b))^*a^*$.

It seems you are almost there. You just need to care substrings with $abb$. One possible way is
$R.E = (a^{*} (ba)^*(bba)^*)^*bbb(a^*(ab)^*(abb)^*)^*$

Note that I came upon this regular expression from the $FM$ of language you described. So, another way to find $R.E$ is from $FM$ directly.