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Conventionally, people will say a probability of zero is equivalent as saying that the event is impossible.

But when we look at the probability from a mathematics perspective, probability is defined as the frequency of occurrence over the how many times the experiment is performed, limit as the number of trials goes to infinity.

Doesn't this mean a probability of zero is an occurrence that is arbitrarily small but possible? What are some of the ways to make this line of argument more rigorous?

Ben Grossmann
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Fraïssé
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    I like to think of it this way: suppose we had a dart with an impossibly fine point so that it would only hit one point inside of the target (treating it as a disk in the plane). The probability of any one point being hit is zero because otherwise if there was a nonzero probability of any point being hit, adding up the probabilities of each point being hit would give infinity. However if I throw the dart, some point must be hit. So yes the notion "arbitrarily small but possible" is likely the closest way to think of it. – Cameron Williams Sep 15 '14 at 23:09
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    Or you can think like the Queen: "Alice laughed: "There's no use trying," she said; "one can't believe impossible things." "I daresay you haven't had much practice," said the Queen. "When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast." $\sim$ Alice in Wonderland – r.e.s. Sep 15 '14 at 23:12
  • The Fourier Transform always bothered me for a similar reason. If you have a signal that is bounded in time, you can prove that it is representable as an infinite sum of sinusoids - each with an amplitude of 0. – Ryan Sep 16 '14 at 01:15
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    "Impossible event" is not part of the terminology of the mathematical theory of probability. In mathematics a probability zero event is a set to which the probability function assigns the value zero. In the real world probability zero means impossible, because there is no such thing as a dart with an infinitely fine point, and there is no such thing as an infinite sequence of trials. – bof Sep 16 '14 at 01:47
  • Theory of infinitesmals, anyone? – keshlam Sep 16 '14 at 05:05
  • @bof: What if I think of a random integer number (no upper or lower limit) and you have to guess it. What's your chance of guessing it in one try? – nikie Sep 16 '14 at 06:39

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One way of making this rigorous is to make an analogue between probability and area. The comparison here is made precise via measure theory, but can be explained without recourse to technical definitions.

Consider two "shapes": a point and the empty shape consisting of no points. Think of these geometric entities as sitting inside two-dimensional space. What is the area of a point? Zero. What is the area of nothing? Also zero. Does that mean that a point is the same as nothing? Of course not. All we can say is that the notion of area is not capable of distinguishing between them.

Using the above example, we can construct two games. In the first game, take a black dartboard and paint a single point red. In the second game, leave the dartboard black. Throw a dart at either one. Is the probability of hitting the red different between these games? Does this mean that these games are equivalent? Does this thought experiment suggest anything about the limitations of the probabilistic method?

Elchanan Solomon
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Many events which mathematically have probability zero are possible. The standard source of examples is random variables with continuous distribution. Here the probability of taking on any given real value is 0, but certainly the variable always takes on some real value! In these cases, with repeated sampling, you will, with probability 1, get a sequence of distinct numbers, so the frequency of the first value will look like $1,1/2,1/3,\dots$.

Ian
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  • Indeed, if X has a continuous distribution on a real interval then ℙ( "X is computable")=0. – r.e.s. Sep 15 '14 at 23:38
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    How so? Any measurement of a property on a real world object results in a value with some truncated error. Thus, the real value isn't what was measured. So over a continuous distribution for example, the probability of a meter stick being 1.00001 meters is effectively 0, and we might measure the meter stick to have that value, but that doesn't mean it actually does. – krb686 Sep 16 '14 at 01:12
  • @krb686 The issue of measuring a number which actually exists precisely in the presence of error is another problem, different from what I'm describing. I'm describing a mathematically idealized "measurement" of a truly random quantity. Such a measurement has infinite precision. One can connect this back to real-world measurements to some extent. For instance, if we are sampling from something which is uniformly distributed on an interval, but we have only finite resolution, then as the resolution improves, the probability of seeing any particular value will tend to zero. – Ian Sep 16 '14 at 01:25
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    (Continued) This example is easy to formalize in fact. The "finite resolution" random variables $X_N$ are uniformly distributed on the discrete set ${ 0,1/N,2/N,\dots,(N-1)/N,1 }$, where each value has probability $1/(N+1)$. As $N \to \infty$ we have that $X_N$ converge in distribution to a variable which is uniform on $[0,1]$. – Ian Sep 16 '14 at 01:35
  • @krb686 - "How so?": The computable reals in any interval form a null set (i.e., whose Lebesgue measure is zero). This is a mathematical issue, not a physical one. – r.e.s. Sep 16 '14 at 04:16
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I have an interesting analogy in mind. I think it might have some value for further understanding. I think about the the problem of reaching from point $A$ to point $B$. The strategy is to take the half of the way at each step between the current position and the target position $B$. If we continue taking the half of the way at each step, we we will have the half way left all the time. But when we take the limit of this strategy the distance between the target $B$ and us will go to zero, meaning that we will eventually reach $B$ but in order to reach we need to take infinitely many number of steps.

Regarding the comment by @Cameron Williams we can hit the target on a say line, not the disk, for some subset (say a closed interval) of this line with some probability $p$. Then an equivalent way regarding the example above suggests that to talk about such a probability for a single point on the line we must try infinitely many number of times, while this probability is defined for any finite number of trial for the subset line (any measuralbe subset of the sample space).

Seyhmus Güngören
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If probability is the number of occurrences, then a probability of zero means there will be zero occurrences. Zero doesn't signify an infinitely small number of occurrences, it signifies no occurrences at all.

Sometimes we don't need an infinite number of experiments to determine that the frequency of occurrence will always be zero. Something as simple as solving a system of linear equations can illustrate the point. If the system is inconsistent then it will have no solution set. You can experiment by plugging in values all the way to infinity and there will never be a solution set.

Whatever the problem, if it can be defined as a system of linear equations and it then solves to an inconsistent system, the solution set is empty. Parallel lines will never cross. In that case the probability is zero.

So probability of zero doesn't mean an infinitely small number of occurrences, it means absolutely no occurrences at all.

Fenris Lloyd
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