By adding constraints such as human lifespan, you change the calculation from an infinite set to a bounded set, which means you are now working with a different problem.

Given a non-zero event probability, as the set size approaches infinity the probability of a single occurrence within the set approaches 1.

Measuring the probability that an event will occur in a single test is different than measuring the probability that an event will occur in a set of tests. Even though the latter depends on the former, you're measuring separate probabilities because the former does not depend on the number of tests performed.

If you consider that other ways of dying prevent you from dying by being hit by a train (which kinda makes sense, you only live die once), then you are not considering anymore the probability of being hit by a train over an infinite period of time. You are just considering the period until you die of some other cause.

On the other hand, if you are some immortal superhero, then yes, in that case, all the other causes of "death" are not final anymore, and in that case the probably you will die being hit by a train will be 1. It absolutely will happen to you. And since you will resurrect right after that, it will even happen multiple times. An infinite number of times. Enjoy!

You are correct. When people say that the probability of a random event approaches 1 over time, it's usually under the assumption that probability is constant. For flipping a coin, assuming a constant probability for "heads" every flip is reasonable. For a person, assuming a constant rate of train-death every year is not reasonable because, as you point out, various things will affect your probability of getting killed by a train, such as "whether you're alive," that change over time.

>Over an infinite period of time the probability of the universe ending is 1, right? Does that not invalidate every mathematical assumption that the probability of [any event] occurring over an infinite span of time is 1.

The first question is a physics question that we don't really have an answer to. Even if it's true, that just means that there will almost certainly be a finite lifespan of the universe, meaning there's no infinite time span for the relevant event anyway. More to the point, math, including probability, will always be at best an approximation of reality. You generally can't assume you'll actually run into infinite anything, but there are situations in which you have so much of something (such as trials of a random event) that you can expect it to behave more or less like the infinite case (such as giving a very, very high probability of eventual success), so doing math about infinite cases can still sometimes serve a practical purpose.

You're absolutely right! There's something called the "Kolmogorov Zero-One Law" that says "the probability of something always happening again is either 100% or 0%". But usually, what people mean by this is that anything where the probability of it happening on any given day is some minimum nonzero amount (or a probability that goes to zero, but slowly enough that the probabilities add up over time), then it's going to happen eventually with probability 1.

This premise, especially given then examples in the body, is not even what we mean when we talk about a probability becoming 1 given infinite time.

In mathemathics, a 0 multiplied by anything, usually only results to 0, so for this fact to be true, there has to be at least some chance of it to happen.

This complete issue with your question being resolved, let's talk about probabilities. When calculating the probabilty of an event X to happen after a given amount of "tries", it is much easier to talk in terms of "how likely is it not to happen." as it is just shorter math to the same result.

When testing twice for a same probability, given you're only trying to get it to happen once, you multiply the odds of it by themselves. So, if something had a 10% chance to happen, it has a 90% chance not to happen, and when you try twice, it's 0.9 x 0.9, which is 0.81. It is lower than the original odds. If you try three times, for it not to happen at all would be 0.9 x 0.9 x 0.9, which is 0.729.

Then, to get the odds of it happening once, you get 1 - [the odds of it not happening]. Using my example, the odds of it happening once in two tries is then 0.19, or in three tries, it would be 0.271.

This trend of it getting lower and lower is what gets the odds of it never happening infinitely closer to 0, and thus the odds of it happening, given it has a chance to happen, AND infinite time, is always virtually 1.

You are confusing thought experiments with reality. A thought experiment is under strict specific assumptions; it does not matter if reality looks different or this cannot actually be.

For example: "if I flip a coin once a second for all eternity, I will get heads at least once"; humans not living for eternity, the coin very slowly ablating away, the sun possibly swallowing Earth some day, or everything being turned into paperclips by a rogue AI don't matter and are not viable scenarios for this. They are idealized away, ignored.

They are usually done to either explore and understand some concept, or to show effects (paradoxical or not) of extreme settings, or are just a kind of game for its own sake. Like a science fiction book.

> Over an infinite period of time the probability of the universe ending is 1, right?

There is no physical reason for this to be so. It might be, but we are definitely not certain about it nor the opposite.

Well...yes. Because if intervening factors occur you're measuring the probability it will happen over the span of time until that intervention occurs, not infinite time.

Basically any time you start hearing "infinite" thrown around you're probably in a purely theoretical scenario with a conspicuous lack of limits or z-factors.