I’ll just explain why I have come to this question here. I was thinking about the multiverse thing. It occurred to me that if the multiverse is real, as in an infinite quantity of universes, then there is an infinite number of universes where you exist in every variation and an infinite number of universes where you don’t exist in every variation. So if, at random, a portal links two universes together, there is a chance that you will link to a universe with another version of you. It seems like the probability of this would he low, but not zero, despite the fact there would be infinitely more universes without a version of you than there would be with one.
In: Mathematics
If there are an infinite number of possibilities, the chance of picking any one particular one at random is 1/infinity.
The probability of choosing any 1 outcome from an infinite span, is the span between the lowest possible non-zero number, and zero itself. While in some sense it’s not zero, it IS zero, effectively. It is a 1 preceded by an infinite number of 0’s, which you can describe *conceptually,* but this can’t actually exist as a realistic number.
Infinity breaks things. A truly infinite set isn’t quite comparable to a finite but immense set.
The chance is “almost surely not” wich is a number that is mathematically indistinguishable from zero, but actually positive. So it CAN happen but you should expect an Infinite number of trials until it happens on average.
https://en.m.wikipedia.org/wiki/Almost_surely
Precisely 0. If we have a continuous bounded set as the event space say reals from 0 to 1 and we ask the probability of picking a given number with uniform prob. distribution (so no bias towards anything) we get 0. Now there are still something to be done here. You can introduce a distribution as asking what is the probability of you picking a value up to some value. Or whats usually more workable is a probability density function which is about the probability of you picking in some interval.
For uniform distribution this would be a constant function, say between 0.25 and 0.75 you have a 0.5 probability of picking form this subset.
It gets tricky with discreet infinite values, infinite discreet sets tend to not be bounded and here the ideas of probability can start to break down. There are of course countable infinite bounded sters like the rationals between 0 and 1 and there are continuum infinite but discreet bounded sets as well like the Cantor set.
It’s zero.
If you have an infinite set and want each element to have the same probability to happen, then:
If the set is countable (i.e. you can count all it’s elements: first, second, thirst, etc), then it’s not possible since any reasonable definition of what it means to sum countable amount of summands will get you “countable sum 0+0+0+… is 0” and “if a>0, then countable sum a+a+a+… is infinite”. Since we want the sum of all probabilities to be 1 (or 100%), that won’t do. So if you want to assign probabilities to a countable set, they can’t all be the same (see two envelopes paradox).
If the set has continuum elements (e.g. dots on a line segment), then you can work with assigning 0 to all elements of this set– if you also add some way of assigning probabilities to at least some of the infinite subsets of this set. The most natural example:
You have a segment of length 1, you pick a dot, and we want all dots be in some sense equally probable. Then you say that probability is 0 for every individual dot, but you also say that probability to pick a dot from a given (sub)segment of length a is a. E.g. the probability that the dot is from the first half of the segment is 1/2 since that half has length 1/2.
It depends what the infinite set of numbers is. If you’re choosing one number among a group that are all ‘equally likely’ (any number between 0 and 1, for example), the probability of choosing one particular number is exactly zero. Zero probability doesn’t mean never for mathematicians, it means “almost never”.
If you’re choosing one number from a list, say {1,2,3,4…}, it is impossible to make them ‘equally likely’. So it would depend on what you assume about your universes. Maybe you think universe 1 has probability 1/2, universe 2 has probability 1/4, universe 3 has probability 1/8, etc. This would be an entirely consistent model, but it has less to do with pure math (odds) than someone’s judgment.
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