In English I always find myself saying "I bet" in this kind of context: "I bet you're well chuffed with it." What would this be in German and could it be used without sounding weird?
There are numerous ways to say it.
Most natural to me sounds (colloquially)
Wetten, dass er sich gleich blamiert? Wanna bet, that he's going to make a fool of himself?
Which is short for
Wollen wir wetten, dass er sich gleich blamiert?
Which can also be used.
As flopana has mentioned, you can also use
Ich wette, er blamiert sich gleich / Ich wette, dass er sich gleich blamiert.
Lastly a famous earlier TV show was also called "Wetten, dass?", where contestants in the show would bet, they could do something extraordinary and famous guests then had to say how they plead. If the guest lost the bet, they often had to do a humiliating task.
I would much rather say ich hoffe, "I hope". Ich wette is closer to betcha and implies a challenge. The expression is optionally negated, unlike I bet, but the conjunctive I wouldn't bet on it works analogously. Ich würde nicht darauf hoffen. The level of politeness may differ, though.
a) I bet he's gonna fall on his face
Ich hoffe er fällt auf die Schnauze
This works well, but the next bit looses in translation.
b) I bet you are well chuffed with it
Ich hoffe, das überrascht dich
I expect you be surprised
Ich erwarte, dass du überrascht bist
Better would be Ich nehme an, "I suppose, I take it that ...", which incidentally covers Wetten annehmen, "to take bets".
As such, "ich wette, du platzt gleich!" would be not only sarcastic, but provocative and thus highly impolite. If you are frequently imposing yourself, then Ich wette damit hast du nicht gerechnet is indeed a fitting translation for the style.
Comparing hoffen to bet makes sense semantically, but etymologically I find hoffen quite difficult (that is I had a counterintuitie hint at a derivation), while bet, that might have come under influence of various lexically not too distant words, is no easier.