photography might be an area where digital hasn’t caught up, since film’s resolution is down to the molecular level

Film resolution is limited by the size of the silver halide crystals that make up the light sensitive layer of the film. Crystals can come in different sizes, but their sensitivity to light depends on their size - generally you need pretty large crystals for usable photographic film, somewhere between 0.1 and 10 microns (depending on the film ISO rating) - about 3-5 orders of magnitude larger than what you would consider molecular scale.

When the film is developed the crystals are visible as film grain limiting the resolution in some ways similar to pixel size of a digital camera (although there are differences, since the crystal size is not completely uniform but rather has a specific distribution, creating a more random effect than the regular pixel grid of digital cameras)

The pixel sizes on modern high resolution digital camera sensors are actually similar, down to 0.5 micron. It’s hard to make an exact comparison, but I have seen estimates that you need a full frame digital sensor of somewhere between 10 to 50 megapixels to equal the resolution of 35mm ISO 100 film.

And modern sensors are much more light sensitive than film, which allows you to shoot more optimally and give you more flexibility (less exposure time, potentially higher f-stop with better lens resolution, lower ISO, less light, etc.) and therefore achieve potentially better results in more conditions. Add to that the hassle and costs of working with film, and most professional photo work is now done in digital as well. Film is generally only used for stylistic purposes, by purists who are not satisfied with digital simulation.

But your citation gives both statements:

“In fact, the monkey would almost surely type every possible finite text an infinite number of times.”

and

“The theorem can be generalized to state that any sequence of events that has a non-zero probability of happening will almost certainly occur an infinite number of times, given an infinite amount of time or a universe that is infinite in size.”

So when you say the number of times is “unknowable” the actual answer is “almost surely an infinite number of times” no ? Since the probability of that can be calculated as 100%. The mindfuck part is that it is still possible that no monkey at all will type a particular text, even though the probability of that is 0.

The probability that only 2 monkeys will type the text is also still 0, same as 3 monkeys, 4 monkeys, etc. - in fact the probability of any specific finite number of monkeys only typing out the text is still 0 - only the probability of an infinite number of monkeys typing it out is 100% (the probabilities of all possible outcomes, even when infinite, have to sum up to 1 after all)

We just know that it will almost surely happen, but that doesn’t mean it will happen an infinite amount of occurrences.

Basically, if we know “it will almost surely happen” then we also know just as surely (p=1) that it will also happen an infinite number of times (but it might also never happen, although with p=0)

Ok, this is interesting, so thanks for pointing me to it. I think it’s still safe to say “almost surely an infinite number of monkeys” as opposed to “almost surely at least one”, since the probability of both cases is still 100% (can their probability even be quantitatively compared ? is one 100% more likely than another 100% in this case ?)

The idea that something with probability of 0 can happen in an infinite set is still a bit of a mindfuck - although I understand why this is necessary (e.g. picking a random marble from an infinite set of marbles where 1 is blue and all others red for example - the probability of picking the blue marble is 0, but it is obviously still possible)

That’s the thing though, infinity isn’t “large” - that is the wrong way to think about it, large implies a size or bounds - infinity is boundless. An infinity can contain an infinite number of other infinities within itself.

Mathematically, if the monkeys are generating truly random sequences of letters, then an infinite number (and not just “at least one”) of them will by definition immediately start typing out Hamlet, and the probability of that is 100% (not “almost surely” edit: I was wrong on this part, 100% here does actually mean “almost surely”, see below). At the same time, every possible finite combination of letters will begin to be typed out as well, including every possible work of literature ever written, past, present or future, and each of those will begin to be typed out each by an infinite number of other monkeys, with 100% probability.

But it also works with infinite number of monkeys, one will almost surely start typing Hamlet right away.

Wouldn’t it even be not just one, but an infinite number of them that would start typing out Hamlet right away ?

The Threnody is definitely his most famous, but he has used that technique in some of his solo compositions for cello as well - example

He’s playing Penderecki