Well no. You can try to count every real number forever and you will miss infinitely many still. Some infinites are larger than others, hence I do not see any reason why “infinite time” would cover “every possibility happening”. On the other hand, if you do have a mathematical proof you could refer to, I would be most grateful.
EDIT: To write out my example, let us consider a machine that picks a random number between 3 and 4 every second. Then there is every second a nonzero chance that this machine (assuming true and not pseudo randomness) will pick, say pi. The range of numbers picked constitute the image of a function from the whole numbers to the real numbers (up to isomporphism), which cannot be surjective. Hence there are numbers not picked even though there was a > 0 chance of picking them every second for an infinite time.
I don’t think I understand your example but I feel like people downvoting you without arguing the math is something that should be left to twitter and reddit.
Then there is every second a nonzero chance that this machine (assuming true and not pseudo randomness) will pick, say pi.
No. The probability of picking any particular number from a uniform distribution is 0.
On the contrary, since the works of Shakespeare are a finite string over a finite alphabet (I can formalize this argument if you want), the probability of typing them out after some fixed large number of keystrokes is some nonzero number 𝑝. With 𝑛 monkeys, the probability that at least one will type out the works is 1 − (1 − 𝑝)ⁿ, which goes to 1 as 𝑛 → ∞.
Now, you are right that this does not mean that the works are guaranteed to be typed out. However, it has probability 1, so it’s mathematically “almost certain”.
I hear what you are saying and agree. I never took the monkey Shakespeare theory seriously. It sounded a bit too poppy and quite honestly the guy that told me was a douche and pronounced giblets wrong. But as a theory you could get anything in a long enough time span and infinite amount of resources. Why or how that matters? Well I just don’t see it.
Of course I am not denying that anything possible could happen. That is contradictory to the assumption it was possible in the first place. What I am saying is just that not all that is possible will happen, even if given an infinite time to do so.
EDIT: Unfortunately, given a setup like this the math says monkey Shakespeare will almost surely happen due to there only being finite variations.
Yep! Relatively speaking almost none of them will be picked. The same is also true even if one had a countable infinite amount of machines trying to pick these numbers.
Well no. You can try to count every real number forever and you will miss infinitely many still. Some infinites are larger than others, hence I do not see any reason why “infinite time” would cover “every possibility happening”. On the other hand, if you do have a mathematical proof you could refer to, I would be most grateful.
EDIT: To write out my example, let us consider a machine that picks a random number between 3 and 4 every second. Then there is every second a nonzero chance that this machine (assuming true and not pseudo randomness) will pick, say pi. The range of numbers picked constitute the image of a function from the whole numbers to the real numbers (up to isomporphism), which cannot be surjective. Hence there are numbers not picked even though there was a > 0 chance of picking them every second for an infinite time.
I don’t think I understand your example but I feel like people downvoting you without arguing the math is something that should be left to twitter and reddit.
Thanks. It was a bit poorly worded, but I do think the original statement is wrong and just wanted to sketch an idea of why.
No. The probability of picking any particular number from a uniform distribution is 0.
On the contrary, since the works of Shakespeare are a finite string over a finite alphabet (I can formalize this argument if you want), the probability of typing them out after some fixed large number of keystrokes is some nonzero number 𝑝. With 𝑛 monkeys, the probability that at least one will type out the works is 1 − (1 − 𝑝)ⁿ, which goes to 1 as 𝑛 → ∞.
Now, you are right that this does not mean that the works are guaranteed to be typed out. However, it has probability 1, so it’s mathematically “almost certain”.
I hear what you are saying and agree. I never took the monkey Shakespeare theory seriously. It sounded a bit too poppy and quite honestly the guy that told me was a douche and pronounced giblets wrong. But as a theory you could get anything in a long enough time span and infinite amount of resources. Why or how that matters? Well I just don’t see it.
Of course I am not denying that anything possible could happen. That is contradictory to the assumption it was possible in the first place. What I am saying is just that not all that is possible will happen, even if given an infinite time to do so.
EDIT: Unfortunately, given a setup like this the math says monkey Shakespeare will almost surely happen due to there only being finite variations.
Oh I get you. I see it the same way. I saw it as an interesting thought experiment.
Is it pronounced like gif?
Even funnier in your example is that the chance of any real number ever being picked is infinitesimally small, instead of guaranteed.
Yep! Relatively speaking almost none of them will be picked. The same is also true even if one had a countable infinite amount of machines trying to pick these numbers.