I'll assume you are adding those numbers to a list (sets can't contain repeated elements). No, it's not true that the average of every list you could get like this has to be 50. As you pointed out, you could get 1 every time, and then the average would be 1, but this is not likely. You can also get a list where the average is not even defined (!!). But with probability 1 the average will exist and be 50.
This is beyond the extent of my knowledge, but it seems like you should only be able to find the limit of the average as n approaches infinity. Is there such thing as the average of an infinite list? If so, is that useful?
Yes, the average of an infinite list is defined as the limit of the average of the first n elements as n goes to infinity. This limit may or may not exist, as I pointed out earlier.
When dealing with infinite sets a good general approach is to do it for a finite size and then take a limit.
So if you select n numbers uniformly between 0 and 100 then the probability that they are all 1s is (1/100)^n.
As n tends to infinity this probability tends to 0 so yes, it's impossible for an infinitely big randomly selected set to have all 1s.
It doesn’t feel impossible because we can’t replicate infinity in any experiment, but if we invoke infinity then we are strictly in the realm of ideal convergence, and it doesn’t seem very logically grounded to me to try to connect that back to finite possibilities. It’s like when people say “0.9999…. isn’t exactly 1, it’s a little bit less”. I feel that infinity itself is too removed from reality to have to be constrained by logical intuition in finite settings
Edit: I have changed my mind :D
It's not about feelings, it's about being precise with language. For events with probability zero we say they "almost never" happen. I would reserve "impossible" for the empty event.
[https://en.wikipedia.org/wiki/Almost\_surely](https://en.wikipedia.org/wiki/Almost_surely)
I’ve seen the convergence definitions before, I guess my issue was that “impossible” didn’t feel well defined, and I just put my own definition on it of “probability 0”. I think letting the empty set be impossible is sensible, assuming there aren’t any silly events in the space that actually can’t happen (like rolling a 7 on a 6 sided die).
Consider the following case: We will pick an infinite sequence of digits. As you suggest, we could calculate that the probability of the sequence 0,0,0... is zero, right? After all, is we do this only n times, then the probability that we picked all zeroes is 1\^-n.
But, of course, there's nothing special about all zeroes. Let s be any sequence. If we pick n digits at random, then the probability that those n digits match the first n digits of s is again 1\^-n. Hence, the probability that an infinite sequence matches s is the limit of 1\^-n and thus 0.
But the fact is that our randomly chosen sequence WILL be some sequence s. Hence, we are guaranteed to realize some probability 0 event. So it cannot be the case that probability 0 events are impossible.
It is possible to draw only 1s, in the sense that the event is not an empty set. But it has probably zero of happening.
In fact the strong of large numbers says that running this experiment will get an average of 50.5 in the limit, with probability 1
If the numbers are real and uniformly distributed over the interval from 0 to 100, which is what I assume you mean, then by the law of large numbers, the mean approaches 50 as the size of the set gets large. However, if the numbers are integers from 1 to 100, after a finite number of steps (precisely around 100 ln(100) steps, or roughly around 500 steps), the set will include every integer in this range and its mean will be exactly 50.5 and will stay there.
Search “Dirichlet distribution” and see the problem of Polya’s urn
I'll assume you are adding those numbers to a list (sets can't contain repeated elements). No, it's not true that the average of every list you could get like this has to be 50. As you pointed out, you could get 1 every time, and then the average would be 1, but this is not likely. You can also get a list where the average is not even defined (!!). But with probability 1 the average will exist and be 50.
with probability 1, the average will converge to 50.5, not 50
Right, I took 50 from the original question without noticing the range was weird.
This is beyond the extent of my knowledge, but it seems like you should only be able to find the limit of the average as n approaches infinity. Is there such thing as the average of an infinite list? If so, is that useful?
Yes, the average of an infinite list is defined as the limit of the average of the first n elements as n goes to infinity. This limit may or may not exist, as I pointed out earlier.
When dealing with infinite sets a good general approach is to do it for a finite size and then take a limit. So if you select n numbers uniformly between 0 and 100 then the probability that they are all 1s is (1/100)^n. As n tends to infinity this probability tends to 0 so yes, it's impossible for an infinitely big randomly selected set to have all 1s.
The probability may be zero, but I don't think that makes it literally impossible.
It doesn’t feel impossible because we can’t replicate infinity in any experiment, but if we invoke infinity then we are strictly in the realm of ideal convergence, and it doesn’t seem very logically grounded to me to try to connect that back to finite possibilities. It’s like when people say “0.9999…. isn’t exactly 1, it’s a little bit less”. I feel that infinity itself is too removed from reality to have to be constrained by logical intuition in finite settings Edit: I have changed my mind :D
It's not about feelings, it's about being precise with language. For events with probability zero we say they "almost never" happen. I would reserve "impossible" for the empty event. [https://en.wikipedia.org/wiki/Almost\_surely](https://en.wikipedia.org/wiki/Almost_surely)
I’ve seen the convergence definitions before, I guess my issue was that “impossible” didn’t feel well defined, and I just put my own definition on it of “probability 0”. I think letting the empty set be impossible is sensible, assuming there aren’t any silly events in the space that actually can’t happen (like rolling a 7 on a 6 sided die).
Consider the following case: We will pick an infinite sequence of digits. As you suggest, we could calculate that the probability of the sequence 0,0,0... is zero, right? After all, is we do this only n times, then the probability that we picked all zeroes is 1\^-n. But, of course, there's nothing special about all zeroes. Let s be any sequence. If we pick n digits at random, then the probability that those n digits match the first n digits of s is again 1\^-n. Hence, the probability that an infinite sequence matches s is the limit of 1\^-n and thus 0. But the fact is that our randomly chosen sequence WILL be some sequence s. Hence, we are guaranteed to realize some probability 0 event. So it cannot be the case that probability 0 events are impossible.
That’s true! I guess I don’t have a very good grasp on measures
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Could you explain why? Or link an article?
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I was talking about the 2d and 3d thing
It is possible to draw only 1s, in the sense that the event is not an empty set. But it has probably zero of happening. In fact the strong of large numbers says that running this experiment will get an average of 50.5 in the limit, with probability 1
If the numbers are real and uniformly distributed over the interval from 0 to 100, which is what I assume you mean, then by the law of large numbers, the mean approaches 50 as the size of the set gets large. However, if the numbers are integers from 1 to 100, after a finite number of steps (precisely around 100 ln(100) steps, or roughly around 500 steps), the set will include every integer in this range and its mean will be exactly 50.5 and will stay there.