The older I become (currently 66) the more I am interested in reading about and learning about the physical universe. It’s reassuring and at the same time humbling to realize the minuscule and irrelevant quality of my individual existence and the impossibility of having any kind of a grasp of ultimate meaning or universal truth. With that tedious preamble, I’ve wondered for years now: given the size and diversity of the universe, is it possible that anything one can imagine is surely happening somewhere, sometime, if not at this moment even including another Steve writing this same question to another Cecil?
IOW: given the size of the universe, what is the probability that a monkey somewhere is typing The Brothers Karamazov as I write this?
No, an infinite universe does not imply that all things occur. For instance, you are probably aware that the set of all integers is infinite - even though this set is infinite, it doesn’t even contain all numbers (such as the sqrt of 2, or 1/3rd, or even 0.5) much less anything else such as you, me, puppies, or kittens. Infinity does not equal everything.
However, there are those who think there may be more universes besides ours, and possibly an infinite number of those universes, and that every possible state can be found in some universe. One theory I’ve heard is that a universe split happnens every time a quantum particle is forced to take one state or the other. That is, when Schroedinger opened the box, it created two universes identical except that one had a live cat and the other had a dead cat.
I think that such theories are untestable so I don’t know how much serious work there is in physics on that.
Probabilities are only defined relative to probability distributions. It’s hard to say that they exist substantively in themselves, though, of course, for various reasons, one would like to try (generally along some sort of frequentist lines). But, really, probabilities can be whatever you want; as a tool for analysis, you can make some assumptions about them, and see where those assumptions lead you, but you’re never going to find yourself discovering absolute facts about what the probability of X is; only “What the probability of X is relative to assumed background probability distribution Y”?
If those finitely many arrangements within a specified volume are all equally likely and independently selected throughout the infinite universe, then the probability that such a monkey-cum-Dostoevsky exists is actually 1. (Indeed, every arrangement will have probability 1 of occurring infinitely often)
(But those are odd background assumptions to make, in this context.)
It says that if the universe is infinite, you can on the average expect to find an exact duplicate of you about 10^(10^29) meters away from you. To find a monkey that’s typing The Brothers Karamazov you would expect to have to search everything out to about that distance, I guess. I pass on defending the statements in this article.