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Is it possible for an event to simply happen for which it is impossible to define any probability?
(Note: By "impossible" I don't mean just "impractical" -- I really mean that the event should not follow any probability distribution.)

Somewhat similarly: can a "random" number generator exist which does not follow any probability distribution?

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    In my measure theory class, the lecturer briefly mentioned how measure theory is the foundation of modern probability theory. The events are measurable sets and their probability is the measure of that set, so perhaps some unnatural type of event corresponds to an unmeasurable set. Hopefully someones answer to this will address this.2011-11-10
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    I think the ''philosophy'' tag is appropriate for this question.2011-11-10
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    The probability that you can find life on Mars (or a Nessie in Loch Ness). I don't think it is "impossible to define any probability", but I think it is mathematically ill-defined (is it?).2011-11-10
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    Perhaps the probability of something is not known, but there is a certain probability that the probability falls within a certain interval. Or a probaility that the probability of the probability of an event is in a certain interval etc.2011-11-10
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    Probably the terminology in mathematics is the reverse of what you are saying. We start with a "sample space" $\Omega$, and then consider only some of the subsets, called "events", and assign probabilities to them. Other subsets, without probabilities, are not called "events". Now of course it is often the case (in mathematical statistics, for example) that we talk of an event such that the probability exists, but we don't know what it is.2011-11-10
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    The probability is given for sets of events, not just a single event. In case of the uniform probability on $[0,1]$, for example, any event (a point) is contained in a non-measurable set. That is, each event is contained in a set for which a probability is not defined.2011-11-10
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    @Mehrdad: it's not clear to me whether this is a question about mathematical probability or probability as it is applied to the real world. Could you clarify?2011-11-10
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    @AndréCaldas: As (@GEdgar) also mentions, events are *sets* that contain points, but the singleton points $\{\omega\}$ may or may not be events depending on the choice of sigma-algebra for which the probability measure is defined. It is *not* true that every *event* is contained in a nonmeasurable set, even in the example you mention. A trivial example of such an event is $\Omega$, i.e., the whole space, and this example holds for all probability spaces $(\Omega, \mathcal F, \mathbb P)$.2011-11-10
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    @cardinal: You are right, my terminology is wrong. Anyways, it does not make any sense to say "an event happen". What "happens" (maybe the word "happen" is ill-defined in the context of measure theory) is a "singleton point". I would change the word "event" to "point" in my comment if I could... :-)2011-11-10
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    @QiaochuYuan: Ah -- I meant it from a mathematical standpoint. i.e. I wasn't looking for something that is *impractical*, but something that's theoretically impossible to give a probability to.2011-11-10
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    If you are talking about the measure theory involved, the concept of an event "happen" is not defined. You have the probability associated with a certain event. But the event do not "happen". Anyway, as I have pointed, what probably you want to discuss about "happening" or "not happening" is not the event, but the single element in it. Actually, you could talk about an event happening... but this would imply that any set (measurable or not) containing this event have also happened!2011-11-11

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