Q: In an infinite universe, does everything that’s possible have to happen somewhere?

The original question was: Lets say that we determine that an event is physically possible. So that means the probability of that event is greater than zero. Right? So my question is this. Is there any sense in saying that the event will NEVER happen even if it has a non-zero probability? In other words, if it can happen, will it happen given enough or infinite time? Does it have to happen eventually?

Physicist: There are a lot of subtleties in this question!  The answer is basically yes, but there are some sneaky assumptions worked into that.

Right off the bat, a probability is always based on “priors”.  For example, “the probability that it will rain today” or “the probability that a 4 will be rolled” are not, completely on their own, well-defined probabilities.

The probability of a particular event is dependent on “priors” or “conditions”.  In order to know the probability of getting a particular result, you must have prior knowledge of what die is being rolled.

Before you can find a probability that’s an actual number, you need to know something about the priors.  The probability that it will rain depends on the place, time, season, whether or not it rained yesterday, etc.  The probability that a 4 will be rolled depends on what kind of die is being rolled, if it’s weighted, or even if dice exist.

In this general case, you may have a tiny, non-zero probability, but if it’s based on priors that are themselves impossible, then the event itself may also be impossible.  You can generalize the priors a lot, but you can never quite get rid of all of them.  For example, it may be possible to prove, beyond a shadow of a doubt, that the probability that a unicorn is violent is 5%, given that unicorns exist (when a scientist says “given”, they’re about to spit out some priors).  However that doesn’t guarantee that a violent unicorn must exist, because it requires that unicorns (with their magic, and virgin sensing powers, and whatnot) must already exist in general.

But (and this may be more at the heart of the question), given an infinitely large universe that’s more or less homogeneous (lots of “stuff”, like the part of the universe we can see with telescopes, instead of just being empty forever), then pretty much anything that’s remotely possible, that could conceivably be the result of a string of remotely possible causes (e.g., horse begets unicorn begets violent unicorn), will happen somewhere.

As a slightly less fantastical example, we can imagine creating, say, an array of Tinker Toys™ linked together and spanning light-years of space.  If this could be instantaneously constructed, somehow teleported into existence, then it would continue to exist for a little while (its self-gravity would start to crush it in fairly short order).  It is not, by itself, an impossible configuration of stuff.

Light years of tinker toys. Its existence is possible, but there is no set of circumstances that would lead to it happening.

However, due to laws like the conservation of mass, the light-speed speed limit, and the nature of gravity, there’s no way to put together a structure this big and massive (or for it to form naturally).  Long before you even got to the business of connecting everything together you’d find that there was already far too much mass, far too close together.  No matter how fast you tried to get everything in place you’d find that the arrangement is smaller than its own Schwarzschild radius, which means that a Tinker Toy™ construction of this size and density is already a black hole (“teleporting it together” gives it a little time because no part of it would “know” that it was too big for a while).

So, we can sit back as say things like, “well, there are X many possible arrangements of atoms, or quantum states, or whatever, and this is one of them…” and can then calculate a ball-park estimate of the probability of this massive wooden grid existing (“1/X” maybe?).  Unfortunately, the answer we’d get would be incorrect, because the priors are messed up.  While it could exist, it could not be formed.  So simply knowing that something is “possible” doesn’t mean that the universe can ever be in a state that would eventually lead to that thing happening.

As far as happening eventually: if it’s not happening now (in an infinite universe), it almost certainly never will.

The tinker toy picture is of a “clockwork” tick-tac-toe playing computer.