How likely/unlikely is an event with probability $1$/$0$?

Question

Probability theory says that if an event $E$ is certain to happen, then $P(E)=1$ which makes sense. Similarly, an impossible event has probability $0$.

What surprised me is the fact that you can still find mathematical texts (notice that this paper comes from a renowned American university) that say the converse are also true, namely:

$P(E)=1 \implies E\quad$ is certain to happen

and

$P(E)=0 \implies E\quad$ can't happen.

Now let's consider the second case. Let's say I'm choosing a point randomly from the interval $[0,2]$. Even though it's possible for every particular point to be chosen (I can easily choose $2$ or $0.5$), the calculated probability for randomly choosing that particular point is $0$. But I have chosen a point, right? Thus it is can happen.

In this case, the probability should be considered as a limiting value. When $P($the randomly chosen number equals $1$$)=0$, it should be understood as the limit of the number of times I've chosen $1$ divided by the number of trials. As the number of trials increases, this fraction approaches $0$ - but it doesn't have to be $0$ at any point during that process.

Similarly, in the first case, I might consider of an event that the randomly selected point from $[0,1]$ is from interval $[0,1)$. The measures of those sets are identical, so the probability equals $1$. Does it mean I will certainly select a point from $[0,1)$? Of course not, because $1$ can be chosen. Thus the event is not certain to happen.

Is there anything wrong with reasoning above? Why are so many people convinced the two implications are true?

Answer 1

It is not true in general that probabilities of $0$ and $1$ necessarily mean that the event is impossible or certain.

Your example with a point chosen from $[0,2]$ shows clearly that such a claim can't be upheld if we want to speak about continuous distributions at all.

---

As Lulu notes in a comment, the text you're linking to contradicts itself: On page 1 it wrongly claims that

Probability always lies between 0 and 1. If probability is equal to 1 then that event is certain to happen and if the probability is 0 then that event will never occur.

whereas on page 3 it contradicts this with the correct

An impossible event has a probability of exactly 0, and a certain event has a probability of 1, but the converses are not always true: probability 0 events are not always impossible, nor probability 1 events certain.

Answer 2

Probability 0, intuitively, means the event is infinitely unlikely to happen: if you wanted to bet you'd hit a particular number in advance, you'd have to be infinitely lucky to win that bet. Infinitely lucky, but not necessarily, impossible.

To understand what I mean by "infinite luck", consider the simple case of drawing a number from the uniform distribution on $[0, 1)$. The drawing process can be imagined by, more or less (i.e. ignoring subtle and nasty little caveats about the non-uniqueness of decimals - that is, the fabled "0.99999..." problem), a (fictitious) "supertask" involving drawing each digit of the fraction (i.e. $0.xxxxxxx....$ representation), one after another one at a time, from a fair 10-sided die (they exist, even ones numbered 0-9 - just do a Google Image Search for "10-sided dice". A 0-9 numbered die is useful in pairs for rolling uniform percentages, where a 100-sided die would be impractical, in certain pen-and-paper role-playing games. You can consider this an incipient form of the great process we are considering here.) one at a time, for an infinitely long time, until the number is settled. One can quite clearly see that if you wanted to draw a specific, intended number - say $\frac{1}{3}$ - you would need to hit the digit "3" every single time you rolled the dice, forever, to the "end" of your supertask. It'd be like buying lottery tickets and always hitting the jackpot, every time, buying them to the literal end of time, as though it were almost law-like, but nonetheless being due to sheer chance. That, intuitively, is both how "unlikely" a probability 0, but not impossible, event is, and how it can be possible despite our intuitions about luck. Remember that there is no reason that at any step the die cannot return a 3, whether we are in 100 digits, a thousand, a million, a billion, googol, googolplex, Graham's number, TREE(3), Rayo's number, etc. . As long as that die is fair it will have probability $\frac{1}{10}$ to give another 3, all the time, so nothing is inconsistent whatsoever about it literally giving 3s until the end of time.

[An impossible event would be like rolling that dice and getting a smiley face instead of a digit, when it has no smiley faces on them (remembering we're talking an ideal universe here, as we always do in mathematics, no uber-unlikely (but probability $>0$) weird quantum tunnelling rearrangement events or whatever). You can roll to the end of time but you are guaranteed to never see that as it's simply not in the offing at all.]

[And if you protest "well surely it has to stop giving 3s!", then thou doth protest too much. It doesn't: if it did, if it became incapable of giving 3s after some point, then that would mean the probabilities changed. In fact the distribution of reals would now no longer be uniform. That's not what an eternal, fair die is. This protest is just another instance of the Gambler's Fallacy, in a rather more profound setting.]

The opposite of these applies for probability 1 with it now being "infinitely unlikely" not to happen.