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Due to this question, I'm wondering about a list of some interesting examples of when the probability that something was going to occur was $0$ and occurs anyways.

I suppose a really basic example could be that the probability that a random number picked between $1$ and $n$ is prime tends to be $0$ as $n\to\infty$, but there are still an infinite amount of primes.

However, I'm interested in less trivial cases (preferably a list) that might very well blow my mind.

Notice: This is not the same as something impossible to occur nor is it the same as something unlikely to occur. Please see Zero probability and impossibility for some explanation.

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Simply Beautiful Art
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  • Not quite sure what you are looking for. By the very definition, an event of probability $0$ can't occur. –  Jan 17 '17 at 01:29
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    @OpenBall That's not true. Consider this answer: http://math.stackexchange.com/a/41108/15140 (Or the answer Simple Art has linked to) – Dair Jan 17 '17 at 01:31
  • @Dair you're right. Anyway, it's already clear from the formal point of view. A set of measure zero is not necessarily empty. I never even gave this a thought. –  Jan 17 '17 at 01:34
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    The chance that someone, asked to pick a random number uniformly between 0 and 1, picks a rational number. The probability is zero, but in practice, it almost always happens. :) – John Hughes Jan 17 '17 at 01:41
  • @JohnHughes Haha, that is true. Darn humans are too rational to be test subjects. – Simply Beautiful Art Jan 17 '17 at 01:42
  • How much rigorous do you expect the answers to be? Are you looking for strictly mathematical answers? I.e. are you just interested in events involving mathematical objects? –  Jan 17 '17 at 01:50
  • @mathbeing Yes please. – Simply Beautiful Art Jan 17 '17 at 01:50
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    @mathbeing You don't need to have rigorous answers or anything. A quick explanation is fine. – Simply Beautiful Art Jan 17 '17 at 01:53
  • In your very own example, the probability to pick a prime number when uniformly choosing an integer between $1$ and $n$ is not zero. It is some value $p_n > 0$, such that $\lim_{n\to\infty} p_n = 0$. This is already not the same thing, and does not fall into the rest of your question (saying that a sequence converges to $0$ is fundamentally different than saying all its terms are zero...) – Clement C. Jan 17 '17 at 02:03
  • @MorganRodgers it's not that they exist, but that there is an infinite amount of primes, despite 0 density in the naturals. – Simply Beautiful Art Jan 17 '17 at 12:22
  • @MorganRodgers It doesn't feel interesting or satisfying, but I suppose this question is too opinion based... I might end up closing it myself D: – Simply Beautiful Art Jan 17 '17 at 15:27

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If you're looking for an example of a nonempty set of measure zero, that's easy: take the set $\{2,5,8\},$ or the Cantor set.

If you're looking for an example of a "real life" event which has probability zero and happens anyway, forget it: probability zero events don't happen. Here are a couple of fake examples:

"Toss a coin an infinite number of times; whatever sequence of heads and tails comes up is a probability zero event."

Nope. In the real world, there is no such thing as an infinite sequence of coin tosses.

"A continuous random variable has to take some real value, but the probability of any real number is zero."

Nope. In real life, a continuous random variable is never observed to take a real number as its value, it is only observed to land in an interval, which has positive measure.

bof
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    Nah, quantum physics says everything is discrete, so no need for intervals. – Simply Beautiful Art Jan 17 '17 at 01:58
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    @SimpleArt That's starting to sound like the most ill-defined, fuzzy question of the day -- and that last comment doesn't help... you're throwing in the sink and the full kitchen. – Clement C. Jan 17 '17 at 02:00
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    @SimpleArt It is still an open problem in quantum physics as to whether there are indivisible units of everything (not just of a given type of matter but also of space or energy). – Ian Jan 17 '17 at 02:03
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    Your examples depend on how one interprets probability. In the subjectivist tradition (De Finetti, Savage, etc.) probability is degree of belief. There's nothing incoherent about my assigning probability 0 to an event that later occurs or could occur. – aduh Jan 17 '17 at 02:10
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    @aduh Good point. In some people's minds, some probability zero events have occurred in recent elections. – bof Jan 17 '17 at 02:20
  • @bof Lmao... I suppose this conversation is my fault. – Simply Beautiful Art Jan 17 '17 at 11:58
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If the probability of an event a random variable is $0$ and the event is surely in the set of possible outcomes, then the only way this is possible is if there are an infinite amount of possibilities.

As it seems to me you are not interested in any cases where there are an infinite amount of outcomes, then nothing "non-trivial" can be found.

Ahmed S. Attaalla
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  • If possible, would you please provide a rigorous explanation? –  Jan 17 '17 at 02:05
  • What would you consider rigorous, usually the probability of an event is taken to mean the number of favorable outcomes divided by the number of total possible outcomes. This number is positive if the number of total outcomes is positive (and finite) and the number of favorable outcomes is positive (and finite), which is given. Since it is strictly positive it can't be $0$. So the only case we would have to consider is infinitely many outcomes/ favorable events. @OpenBall – Ahmed S. Attaalla Jan 17 '17 at 02:21
  • Now I see, thanks. –  Jan 17 '17 at 02:24
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There is no way to choose a random integer with a probability distribution uniform on the integers: the probability of any particular integer will be $0$.

Is getting a random integer even possible?

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Ethan Bolker
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  • I think this is more along the lines of something trivial relating to the fact that there is an infinite amount of natural numbers. I'm looking for something less trivial... – Simply Beautiful Art Jan 17 '17 at 01:51
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Maybe an interesting example you have not seen: a Wiener process is a continuous version of a random walk. If I generate a Wiener process for $T$ seconds then the probability of generating exactly this process is 0.

Matt
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  • I think this is more along the lines of something trivial relating to the uncountability of the reals. – Simply Beautiful Art Jan 17 '17 at 01:50
  • Yes of course this is resting on the same arguments. My point is that this example is somehow "more natural" to me than existence of measure 0 sets on the real line. Wiener processes are something which you may encounter in models of real world phenomena. Just my (biased) opinion, though! – Matt Jan 17 '17 at 01:53
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    To be honest, I don't see how this is more trivial than the example with the primes. – Clement C. Jan 17 '17 at 01:53
  • @ClementC. Hmm... I suppose my example was motivated by trying to provide a "real world, interesting example." I did not consider the triviality of the statements. I suppose an interesting question here is what measure I am using on my function space... – Matt Jan 17 '17 at 02:06
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    @Matt I believe you are reading my comment the wrong way. The comparison I made between your example and the OP's was the other way around. – Clement C. Jan 17 '17 at 02:09
  • @ClementC. Indeed! :) – Matt Jan 17 '17 at 02:12
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A paradox may provide an example.

The unexpected hanging paradox (described here, for example) describes a process for determining that an event has zero probability of occurring, yet the event still occurs.

This, and similar paradoxes, doesn't require an infinite range of probabilities.

But given the self-contradictory nature of your premise, I doubt that you will discover any example that isn't paradoxical.

Jim
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