Improbable vs Impossible? I was wondering how mathematics in general or any of its sub fields e.g.statistics, probability, define the words Improbable and Impossible. 

I get their English meaning, that something is impossible means it is never going to happen and improbable means something is unlikely to happen.

Could someone please provide a mathematical description for these two words as used (if they are used) in some field of mathematics? May be in terms of size of the probabilities. 

Thanks.
improbable = something that can happen, but its probability is comparatively low, but not zero. The distinction between probable and improbable is not, as far as I know, exactly defined.

if you roll one hundred dice, it is very improbable that all of them will land as sixes.

almost impossible = something that can happen, but its probability is exactly zero

if you roll a frictionless die, it is almost impossible that it will end up on its edge.

if you keep rolling a die until a six lands, it is almost impossible that you will never stop rolling.

impossible = something that cannot happen.

if you roll one hundred (standard) dice, it is impossible that at least one of them will land as a seven.

probable

if you do get one hundred sixes in a row, it is probable that the casino will kick you out.

almost definite

if you attempt to balance a frictionless die on its edge, it will almost definitely topple to one of its faces

definite

I think you will definitely try to balance a die on its edge. If you do succeed - remember it's due to friction.
Mathematically an event E can be called impossible if and only if $Pr(E)=0$.  I don't believe we can define improbable statically (i.e. in a way that doesn't change based on context).  One attempt, however, might be: given a threshold probability of $p_T$, any event $X$ is considered improbable if $Pr(X) \lt p_T$.