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?
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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 $1,1/2,1/3,\dots$.
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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).
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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.
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?