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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.

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    $\begingroup$ I think its used the same as in the English language $\endgroup$
    – user93089
    Oct 5, 2013 at 14:46
  • $\begingroup$ I'd say that "improbable" means "with probability zero" and "impossible" means "leading to contradiction, yielding false result". Let a random variable $x$ follow a uniform distribution on $[0,1]$. I'd say that getting $x=0.2$ is improbable, and getting $x=2$ impossible. $\endgroup$ Aug 4, 2016 at 22:15

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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.

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  • $\begingroup$ 'almost impossible' is not 'exactly zero' in any natural or informal language; that goes against any useful definition of 'almost'. $\endgroup$
    – Mitch
    Oct 5, 2013 at 15:26
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    $\begingroup$ @Mitch but, we need some term for "possible, but with zero probability", and "almost impossible" is what stuck to be what is defined as such. Also, see the link I've edited in. $\endgroup$ Oct 5, 2013 at 15:28
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The standard framework of probability theory attempts to assign to each outcome $X$ a number $P(X)$ between $0$ and $1$, which we call the probability of the outcome. The higher the number, the more likely that outcome is to occur. Depending on context, a sufficiently small value of $P(X)$ will correspond to something being improbable, but there's no inherent threshold between improbable and likely.

General probability theory does not have a good way of making sense of "impossible" or "necessity". Having a probability of $0$ does not mean something cannot happen, and having a probability of $1$ does not mean that something must happen. To explain this with a metaphor from geometry, consider the notion of area. The area of nothing is $0$, but so is the area of a single point. In that sense, the notion of area cannot distinguish between nothing and that which is "infinitely small".

Similarly, when trying to applying probability theory to problems that have infinitely many outcomes, the probability of an event can be zero even if it is possible, so long as there are infinitely many other possibilities that are just as likely (or some similar situation). When it comes down to it, the real number line does not have infinitely small numbers, and since probability theory uses real numbers, these events can only be assigned a probability of $0$.

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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 $Pr(X) \lt p_T$.

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    $\begingroup$ incorrect - zero probability != impossible. $\endgroup$ Oct 5, 2013 at 15:36
  • $\begingroup$ I may be inclined to agree with you. Are you saying that if in a sample of size $n$ we do not observe a given event $E$ to happen at all then this doesn't necessarily mean the event is impossible? Or are you saying that given a discrete or continuous distribution which specifies $P(E) = 0$ this event is still possible? $\endgroup$
    – Patrick
    Oct 5, 2013 at 16:36
  • $\begingroup$ It's certainly possible with a continuous distribution. The probability of every single point on a real line interval is zero, but any single point is possible. As for the continuous distribution: let E = 1 if some condition is satisfied, and 0 otherwise. Then E = 1 is possible iff the condition is possible, and E = 1 has zero probability iff the condition being satisfied has zero probability. $\endgroup$ Oct 5, 2013 at 16:43

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