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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!

7 Answers 7

-1

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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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.

  • 0
    I like the first paragraph, it defines what exactly P(A) = 0 is.2014-08-14
25

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.

  • 1
    @lkessler What, 1/infinity being an infinitesimal is like the definition of an infinitesimal. https://en.wikipedia.org/wiki/Infinitesimal2017-06-21
13

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.)

  • 0
    @Doug: that's an interesting question. I suppose one could write down a generalization of probability measures that are allowed to take value in the nonstandard reals and perhaps such a theory would allow such things.2011-06-13
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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 $\mu^*(A) = 0$

  • 0
    When you throw a dice, you will never get "nothing" or something like "both 1 and 2". I think the former is impossible and the later improbable?2014-05-25
11

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.

  • 0
    So I guess this means the notion of impossibility is external to the 'probabilistic way of thinking' (as it's described [here](https://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/)) since whether an event is empty depends on the particular representation (i.e., probability space) used?2016-01-08
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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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    @Tim: I mean that probability theory *can* model the real world (as far as we know), but that it can do more if we want it to.2011-05-27