Can an observed event in fact be of zero probability?
Of course, I know that there exist non-empty events of zero probability. Mu question is the reverse: given that we have observed an event (and we have no other information about it, just the fact that it has been observed), is it possible that the event has in fact zero probability? Or does an observation necessarily mean that the probability is strictly positive?
Example context for the question:
Suppose $x_i$ (countably many) are i.i.d on $[0,1]$, but we do not know the distribution they come from. It may be uniform on $[0,1]$, may be discrete, may be any legitimate distribution (discrete or continuous). Just imagine we have some sort of a machine that shows us one by one a list of randomly drawn numbers between $0$ and $1$. We are comparing the observed numbers one by one to some special number chosen beforehand, for example $\frac12$. Now, given that at some iteration we have observed that special number at least once, does that mean that $\frac12$ has some positive probability under that (unknown) distribution? And if the probability can be zero, can we nevertheless say that we will necessarily observe $\frac12$ again later, if we continue the experiment ad infimum?
Also, disregard the "real world limitations" such as an inability to produce truly uniformly distibuted numbers, or rounding errors or any such thing.
Best Answer
Yes. For a situation where this always happens, assume that one observes a random number $x$ drawn from the uniform distribution on $(0,1)$. Then the probability to observe $x$ is zero.
(Proof: For every interval $I\subseteq(0,1)$ the probability to observe a number in $I$ is the length of $I$. For every positive $\varepsilon$, there are intervals $I\subseteq(0,1)$ which contain $x$ and have length less than $\varepsilon$ hence the probability to observe exactly $x$ is less than $\varepsilon$, for every positive $\varepsilon$, QED.)
Edit: In a discrete space $\{x_i\mid i\in\mathbb N\}$, the probability to observe $x_i$ is, by hypothesis, some positive $p_i$ hence the above applies only to continuous distributions (or at least, partially continuous).