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I read a comment under this question:

There are plenty of events that can occur that have zero probability.

This reminds me that I have seen similar saying before elsewhere, and have never been able to make sense out of it. So I was wondering

1. if zero probability and impossibility mean the same?
2. if an event with zero probability doesn't mean that the event is impossible to occur, how probability theory represents/describes impossibility?

Thanks and regards!

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8 Answers

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Two Schools

I think the crux of the matter is what probability actually is:

• The Bayesian view - probabilities are measures of (personal) confidence or belief, so it's quite obvious why an event with probability zero is not the same thing as an impossible event. But perhaps this isn't such a satisfactory answer.
• The frequentist view - probabilities are the asymptotic frequency of events as the number of independent trials tends to infinity. Here again wee see that something that happens with probability zero is not the same as something impossible; it's just something that happens so infrequently that the numerator in $\dfrac{\text{occurences}}{\text{trials}}$ is dominated by the denominator.

Technically Speaking

Putting aside such philosophical matters, there's also a technical matter to be discussed here. Under the usual measure-theoretic formulation of probability theory, we have a sample space $\Omega$ and a family $\mathcal{F} \subseteq \mathcal{P}(\Omega)$ of events (measurable subsets of $\Omega$), and the probability of an event $A \in \mathcal{F}$ is its measure $\mathbb{P}(A)$. There is nothing in the axioms of measure theory which say that a non-empty set must have a non-zero measure; and if we interpret $\mathcal{F}$ as the set of all possible events, it's clear that an impossible event is not the same thing as an event of zero probability.

Example

To give a concrete example, consider a random variable $X$ which is uniformly distributed on the interval $[0, 1]$. Although $\mathbb{P}[X \in (a, b)] = b - a$ for all $(a, b) \subset [0, 1]$, the axioms of probability force us to conclude that $\mathbb{P}[X = x] = 0$ for any individual $x \in [0, 1]$: for if $\mathbb{P}[X = x] = \varepsilon > 0$, because $X$ is uniformly distributed, by additivity of the probabilities of disjoint events, we'd be forced to conclude that $[0, 1]$ contains at most $\frac{1}{\varepsilon}$ (a finite number!) points, which is absurd.

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Zero probability isn't impossibility. If you were to choose a random number from the real line, 1 has zero probability of being chosen, but still it's possible to choose 1.

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Adding to what others have already mentioned. There is also this notion of plausible event. I am not sure if this is standard. But in the book "Measure Theory and Probability" by Malcolm Ritchie Adams and V. Guillemin, a plausible event is defined as an event which corresponds to a Borel set.

Hence, my understanding of the three words is as follows:

If we take the probability space $(X,\mathscr{F},\mu )$,

An event $A \subseteq X$ is impossible if $A = \emptyset$

An event $A \subseteq X$ is implausible if $A \notin \mathscr{F}$

An event $A \subseteq X$ is improbable if $\mu^*(A) = 0$

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Mathematicians generally formalize probability using the notion of a probability space and measure theory. In this formalism it is possible for an event to have probability $0$ without being the empty event. Perhaps the simplest "realistic" (and I use the word loosely) example of such an event is the event of flipping only heads infinitely many times. This event has probability $0$, but it is not empty, which is what one might call a formal definition of "impossible."

The underlying probability space is the set of possible ways to flip a coin infinitely many times. An example of an impossible event here is that you flip, say, cat. The coin has only a heads side and a tails side; it doesn't have a cat side, so flipping cat is impossible.

(Whether this formalism says anything reasonable about the real world is debatable. In practice, events of sufficiently small probability are already impossible. The above is just a statement about a certain mathematical formalism that has proven to be useful in certain contexts. In mathematics, we want to prove statements about some class of objects. Sometimes we can prove that the statement holds with probability $1$, but this does not imply that it holds for all objects, and since we actually care about all objects this distinction really does need to be made in mathematics.)

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Let $A$ be an event, $\Pr$ be the probability measure.

$A$ has zero probability if $\Pr(A) = 0$.

$A$ is impossible if $A=\emptyset$.

Impossibility implies zero probability, but the reverse is false. Consider the real line $\mathbb{R}$; if you randomly select a number $x$, the probability that $x=0$ is $0$, but this is not impossible. In fact, the probability that $x$ belongs to some countable set, e.g $\mathbb{Q}$, is also $0$.

From a purely mathematical point of view, impossibility is simply a stronger statement, so impossibility cannot be described by probability measure. However, another way of thinking might shed some light. That is, if the probability that something exists has probability greater than $0$, then it exists. This notion has been used for some mathematical arguments.

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Probability theory is an abstract subject, which is not limited to the real world. In cases where it is limited to the real world, an event of zero probability will not occur. But the abstract underpinning of the real-world cases allows for the occurrence of zero-probability events; when you translate these abstract events into events that are physically detectable, their probabilities become non-zero.

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I believe the root of this confusion is all because Math(in its nature) is in general(including probabilities) always more serious towards its concepts, than we -the humans- are. And this seriousness is exactly what brings Math its clarificationability(the power of turning vague into non-vague) and simplificationability(the power of turning complicated into simple), two of its core ultimate goals!

An instance of this amount of seriousness: When Math says "probability" it really means "probability"! I.e, mathematical probabilities are ALWAYS about "probable" expectations, and so even its 1 is not certainty, and even its 0 is not impossibility! Rather indeed, its 1 is absolute probability and its 0 is absolute improbability!

It's just an intentional neglection of all the odd possibilities. For instance, when I throw a die, it is possible that I will get a 7...! Why?! Maybe someone has added a dot to the 6 side! But we neglect that possibility on purpose for the sake of simplification, because it's an odd(read improbable) possibility.

Another Example: Yes, it is possible (even absolutely possible!) that the exact number 3 might occur in the continuous interval [1, 4], but it is absolutely improbable(notice how you can't disagree), thus the P=0. (Notice: Computers are never really continuous, so don't even try doing it programmatically, it won't be a counterexample).

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I don't know if this is correct, but this is how I made sense of it, I think this a very intuitive explanation.

Imagine a car moving (from left to right) in a straight line with constant speed of 100MPH, and at x=0 the breaks of the car are engaged. Naturally the car will start to decelerate until it comes to rest (speed = 0 MPH).

The car will certainly come to rest, but the question is:

At what distance D [meters] (with respect to x=0) will the car come to rest? (This value D is not known apriori and can vary from one experiment to the next by a myriad of factors, so we can ask probability questions about the variable D).

Since distance is modeled by real numbers, the variable D is continuous. If we ask: what is P(D=6.354), the probability that it comes to rest at specifically D = 6.354 meters?

then, since the probability distribution of D is continuous, the answer is P(D=6.354)=0. This probability is zero.

Notice that there is nothing special about the number "6.354". That is, for any real number r the probability that D is exactly r is zero, i.e. P(D=r) = 0. This is simply a consequence of how continuous probability distributions are defined.

Regardless, we know that the car will certainly come to rest and that this happens at some specific distance d (where d is a real number). So even though P(D=d) = 0, it does not mean that coming to rest at D=d was impossible; clearly it was not because the car does come to rest at d.

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