I think you are getting confused by two allusions that happen to share the words "on the wall", but which have nothing to do with each other.
"The handwriting on the wall" is a reference to the book of Daniel in the Bible. According to this book, the imminent doom of the city of Babylon was predicted by handwriting that miraculously appeared on the wall during a drunken party, and which the prophet Daniel then interpreted to mean that Babylon would soon fall. (Some commentators say that the handwriting was not miraculous in any way but was simply ordinary graffiti, to which the king and Daniel attached prophetic significance. But whatever, not here to debate theological questions, just to explain the reference.)
Note that the reference is unlikely to be recognized without mentioning the handwriting. It's not just something on the wall, it's the handwriting on the wall. And it's not really a discussion of probability -- maybe you were just struggling for words there -- but a warning of impending disaster. No one uses this phrase to refer to something good, but only to something bad. And it is normally understood to mean not a probability but a certainty.
"Liar, liar on the wall" and "teacher, teacher on the wall" are references to the fairy tell of Snow White. In the fairy tale, the evil queen has a magic mirror. She regularly asks this mirror, "Mirror, mirror on the wall, who's the fairest one of all?" "Fairest" here meaning "prettiest". As long as the mirror assures here that she is the prettiest woman in the kingdom, she is satisfied. But when the mirror tells her that Snow White is prettier than she is, she becomes furiously jealous and orders Snow White to be killed.