What is the probability of something happening given that time always will exist? If time never ends, is it 100% likely that I will be born again after death?
I have a great way to illustrate my question:
Visualise a die with an amount of sides equal to all possible positions of all atoms. If you roll the die an infinite amount of times, the probability of a given state must be 100%. But if some atoms cannot possibly be at a position, the probability cannot be 100%.
Have a look at the idea of the Boltzmann Brain. Cosmologists are starting to apply the same approach using this idea, as Bostrom used in his version of the simulation argument. You can see an accessible presentation on the controversy about whether Boltzmann brains exist here. This is the context of you 'popping into existence', memories fully formed, in a vast cosmological future. On balance, when you run the numbers, these seem deeply unlikely - the gas in a 1 litre bottle won't randomly all be on one side of the bottle once in the 14 billion years of our universe, so these complex structures without 'causes', seem very unlikely.
'You' in an ontological sense, are your causes and conditions, past and future, and they exist in a vastly complex web which requires every element of the universe that has affected you to be the same. That involves enormously compound unlikely quantum events. So in an ontological sense 'you' in the fullest sense of not only small variations of what actually happens to you, but all your realisable possibilities in this moment, can be recreated in the multiverse, but only very rarely (in the sense of the total probability space, called the Hilbert space), ie when a big chunk of an alternate universe matches ours.
Another framing is, all the possibilities that are realisable from the moments of your life exist in the multiverse, but are forever separated and diverged from the 'you' you consider yourself to be, and not causally linked in any way. You might say there is a branching structure of people that resemble you in different ways, fulfilling the possibilities present in your life (but not impossible things, that's Boltzmann brain territory). I would say alternate 'you's are not meaningfully connected to you, except by you imagining them, and shaping your future actions by learning from that. When you diverge, you cease to 'be' them (see Teletransportation paradox)
In terms of reoccurring in our own universe. Our experiences don't go down to quantum accuracy. It's plausible, likely even according to Bostrom, that your brain and life will be simulated in the future. This can only be done to a certain 'resolution', and it's largely a matter of definitions whether simulations are 'really you'. You could be a simulation now and not know it, which would meet a kind of 'Turing test' analogue, the 'walks like a duck..' test, if you will, for 'really you'. It's hard to draw conclusions about how often these might run and h9w much variation, but thus is really the only way alternate individuals that are meaningfully 'you' could exist.
Probability is slipperier than you imply. The idea all probabilities add up to 1 is called the axiom of 'unit measure' in probability theory, or 'unitarity' in physics. But consider the dice - what about the one time where it balances on a corner or an edge, perhaps have created a pit in the surface from near-infinite rolls? What if the dice has so many sides many of them will never be rolled in the age of the universe, how could you check it's unitarity?
Unitarity is a fiction, a useful and intuitive one, based on imagining counterfactuals: if everything was held to the same initial conditions and repeated infinite times, what fraction would be of each result? We apply limit theory, to move from practical numbers of rolls to mathematically imagining 'as rolls approach infinity'. But in the end, the universe does what it does, and has the final word. Probability is only us imagining 'what if?' and applying simplifying assumptions based on our understanding of a system, invariably limited to simple systems, or else being very unreliable probabilities! (eg. election results)
In quantum mechanics things start to really stretch our understanding. Dirac introduced the idea of using negative probabilities and negative energies in calculations, and predicted the existence of antimatter from this. Feynman argued they need not be thought of as 'real', but as useful book-keeping. And the extended probability ensemble goes further, allowing both negative and beyond unitary probabilities.
Additional assumptions beyond infinite time are needed. For example: imagine a toy model universe, with 3 spatial dimensions, where there are a finite number of particles that begin close together, and where the only law in this universe is some simple law of repulsion. In such a universe the particles will fly apart from each other further and further as time goes on, and will never attract or coalesce, even given infinite amount of time. Our actual universe may in fact be expanding and so may be somewhat similar to this toy model "globally". But even if it's not the point remains that you need additional assumptions.
For example, with your dice analogy the additional assumptions (implicitly assumed properties of dice rolls) are that there are a finite number of states (sides of a die), and that the probability of each state occurring is independent of what has already happened. This guarantees that the probability of any sequence of dice rolls occurring will approach 1 as the number of rolls approaches infinity.
It is impossible to destroy and create matter. Thus, everything that has existed still exists, and everything that will exist already exists, just not in the state(s) it did or is going to. And even if time is infinite you would not be born again. An exact copy of you might have existed or might exist in the future, but not exactly you.
I fear that the questioner and the other answers are falling for the gambler's fallacy
OP, you seem to be in your mind doing an equation that {[lim (1/n)*n as n-> infinity] = 1} and concluding that the probability of anything happening in an infinite timescale also = 1.
To use your die with infinite sides analogy, once the fates roll the die, each next roll is independent. there is no "they have not rolled 16 yet, so 16 is due." Each time they roll the die, it is a new, independent roll.
Imagine a universe that consists only of a counter that increments by 1 every second. In the sense of probability that you describe, every integer has a 100% chance of appearing on the counter because both sets are countably infinite. If the set of all configurations of matter and energy is finite or countably infinite, then I think you are right and Boltzman brains exist. If space is uncountably infinite or divisible into an uncountably infinite number of segments then things get tricky. My guess is that if both time and space are infinitely divisible into uncountable sets of segments, the necessity of Boltzman brains depends on whether the sets map onto one another 1:1. If there are more configurations of matter than segments of time (which I suspect is the case due to extra degrees of freedom) then they are not necessary.
It depends on how the universe ends. Watch this video: https://www.youtube.com/watch?v=4_aOIA-vyBo
Big rip/heat death: The answer is no. The entropy increases to maximum after a finite amount of time. So, even if time is infinite, after heat death, universe just stays in this high entropy state, without formation of new planets and life (which includes you).
Big crunch & Big bounce: The answer is yes. If there is infinite amount of universes in series, there is a probability of 1 for anything to happen again (in bounds of physical laws), which includes you.
