about article

[alan]

hi ,everyone and teachers!
it has been a long time since i went to here before.

i wanna ask ...
what is the difference between
"the possibility of xxxxxxxxxxxx...." and "a possibility of xxxxxxxxxx.." ??
likewise,between "the opportunity to do sth........" and "an opportunity to sth...." ???

i wonder how i can distinguish them ..?
i know that "the" is a definite article,so i think it only makes sense to use "the" instead of "a/an" in the above-mentioned examples,cuz we are referring the possibility or opportunity about sth particulary !
am i right???
If not,please provide me with some examples,thx.

[Tdol]

The possibility= specific- the possibility of X winning an election.

A possibility= one, possibly of more- there's a possibility of rain tomorrow.