It might be a duplicate but what's the difference between almost surely and surely?
I have heard that the concept of "almost" is relatively new in probability theory, but I never understood the core concept that differences it.
From what I have heard, it's linked to the measure in the probability tribu that defines a probability function, but it doesn't help me much.
Is "surely" even defined with new probability theory? Why did they added an extra word?
In this article, Wikipedia states that : "In probability experiments on a finite sample space, there is no difference between almost surely and surely. However, the distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability zero."
I don't understand this sentence. Maybe someone can help me understand this concept and maybe provide an example?
Best Answer
The easiest way to see the problem is by considering an uniform distribution on the interval $[0,1]$. Obviously the probability that drawing a random number $x$ gives a real number from $[0,1]$ is $1$. But given a real number $r\in[0,1]$, what is the probability that $x\in\{r\}$, that is, $x=r$?
Well, it has to be the same for all $r\in[0,1]$, or else we'd not have an uniform distribution. But consider if it has any value $p>0$. Then there exists a natural number $n$ such that $np>1$. Now take a set of $n$ different numbers, $X=\{r_1,r_2,\ldots,r_n\}\subset [0,1]$. What is the probability to draw a number from that set? Well, according to the rules of probability, it would have to be $np$, but we just established that this value is larger than $1$, which cannot be!
So the only probability we can assign to the set $\{r\}$ without getting any inconsistency is the probability $0$. But then, if we assume that probability $0$ means an impossible result, then we just have claimed that for any number $r\in[0,1]$ it is impossible to draw that number! But that would logically mean that it is impossible that the number we draw in in $[0,1]$, which is in direct contradiction to the original assumption that we get a number from that interval with certainty.
The only way to evade that contradiction is to accept that probability zero does not mean the event is impossible. But what, then, does probability zero mean?
Well, it means that almost certainly you'll lose if you bet on that event. That is, your expected win will be negative no matter how little you bet on it, or how much you'll win if the event actually occurs. An event of probability $0$ is an event that is always a bad idea to bet on, even though it may occur in principle.
Note that of course it is also a bad idea and a sure loss to bet on an event that is actually impossible to happen, so this is consistent with the rule that impossible events get the probability $0$.