Events with non-zero probability but with very low probability (say, one in a quintillion) don't happen much. But if you had, say, a quintillion regions for that event to happen (assuming it's a region-wide event that requires a region to occur), then it's much likelier to occur. Further, if you have one hundred quintillion regions, it's likely to happen one hundred times. So, if you have an infinite amount of space, it's likely to happen an infinite amount of times--as is any event with non-zero probability.

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So then in an infinite environment, probability does not exist because everything that can occur has a 100% chance of occurring infinite times, and anything that does not occur is simply impossible via the very nature of infinity?

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Or so it would seem.  But what if each planet is completely random and there are an infinite number of of possible planets.  Then, given an infinite number of planets what are the odds that you'll get a planet that's exactly the same as a randomly chosen planet.  I unfortunately have no idea as to what to do with such a situation.  

I suppose whether or not you're guaranteed to find an identical planet depends on "how infinite" the number of planets there are is compared to "how infinite" the number of possible planets is would determine this, but beyond that I don't know.  I suspect it would be some kind of set theory problem (the cardinality of the set of all possible planets compared to that of the set of planets in the universe?) but it seems vaguely reminiscent of some basic calculus involving infinite limits and growth rates.  I don't know because I don't even know how you'd begin to define that type of infinity.

It's interesting to think about though because who knows if, given an infinite of time, the universe is bound to be in exactly the same state that it's in now?

In the case of your original post though I still contend that it's impossible to tell simply because there's no reason to assume that the probability of their being a planet outside of what you can observe is anything other than just plain zero, and if it's zero then no matter how infinite the amount of space is another planet will never occur.

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However, if there is any reason to believe it is nonzero, given infinite time passage, it's all infinite. Set theory would be relevant in a determinate time frame.

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I guess the fact that it's a quantum universe means that it's possible for a planet to exist outside the observable area, but that also means that in a finite time frame there would be a finite, if small, probability of a planet coming into being as well.

Regarding my last post I we're all going to live forever in some sense.   There must be a range of states of matter that allow for us to be our own conscious selves (if that even means anything at all), because our bodies vary in physical makeup as we live, and so the probability of matter existing in that same state some time again after our death is greater than zero.  Given an infinite amount of time we're all bound to be back sometime.  

Although the fact that this also means that an infinite number of times we're also going to come in to existence in conditions far less favorable than the ones here is not a particularly comforting one.

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