what is the probability of any arbitrary event in infinity? I think it should be certain, or otherwise, we didn’t reach infinity yet (by definition)

In: Mathematics

Consider 1/3 which has a decimal representation of 0.333…. the 3s go on to infinity.

The probability of a given digit being a 2 is 0 though.

So even though we have an infinite amount of digits some digits never appear.

The parameters of a question have to be set up very formally – we must be very precise about what we’re asking – in order for math to have a firm answer for this.

Are you asking about the chance of something happening “given infinite tries?” Sometimes people ask what are the odds that somethinglike a gold watch just randomly assembles itself somewhere spontaneously in the universe, if we assume it’s an infinitely big place.

Or are you asking about the odds of a specific event in a *distribution* of infinitely many possible outcomes? This type of question might be expressed like: what are the odds of a dart landing at exactly (x,y) if thrown randomly at this area?

This is a paradox pretty well-explained in quantum physics… All potentials exist simultaneously, but can only be measured once observed for obvious reasons. Time is linear, and we are time-bound beings. We can only view and measure so many potentials as our technology and the laws of the physical realm within which we exist will allow.

The answer to your question which I perceive to be “what is the probability that some event will occur given infinite time” is, p = 1.

If you really want to dive deep into the fundamentals, in mathematics, what you’re referring to is “[Convergence of Random Variables](https://en.m.wikipedia.org/wiki/Convergence_of_random_variables)”.

I think this is an interesting question and I’m curious what others say.

I would posit that all events won’t happen. Let’s consider a game of heads and tails. You can imagine all finite sequences (an “event”) should occur (such as 1000 tails in a row). But what are the odds of infinite tails? That’s a possible permutation but does not seem guaranteed to occur.