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.
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$\begingroup$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.
$\endgroup$ 2 $\begingroup$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 $0$.
$\endgroup$ $\begingroup$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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