Math Help - Interpretation of Probability

1. Interpretation of Probability

What interpretation of probability is used when one says that the $P(E) = 0$ does not imply that $E$ will not happen, where $E \in \Gamma$ ( $\Gamma$ is some sample space and $E$ is an event in it).

If we used the relative frequency definition, then this statement would be false.

2. Originally Posted by Sampras
What interpretation of probability is used when one says that the $P(E) = 0$ does not imply that $E$ will not happen, where $E \in \Gamma$ ( $\Gamma$ is some sample space and $E$ is an event in it).

If we used the relative frequency definition, then this statement would be false.
Note that this question is only raised for uncountable probability spaces: otherwise, $P(A)=0$ iff $P(\{a\})=0$ for all $a\in A$, hence $A=\emptyset$ if we exclude (as is natural here) the case when some point has zero probability.

In particular, the question refers to problems dealing with the infinity, hence the notion of "happening", which evokes the reality of an event, is unclear. The answer may depend on what is considered "possible": is it possible to play heads and tails and get heads indefinitely? This is "infinitely unlikely" but conceivable.

These unempty events with zero probability are an abstraction, but often make a more concrete sense as a limit: the above example has zero probability because the probability of getting $n$ heads in a row converges to 0 as $n$ grows to infinity. This way, only finite spaces are involved. Of course, this is just a way to hide the infinity under the carpet.

Anyway, I may be wrong but I think that the interpretations of probability (like the one you are evoking) mostly refer to finite probability spaces.