If I remember correctly, Brendan Sullivan had a reputation as a TA for Concepts of Mathematics at CMU as "Math Jesus", not sure if that was a testament to his pedagogical skills or just due to the long hair and beard...
This stuck out to me too, glad someone else noticed it. Just in case people aren't aware, the law of large numbers is a phenomenon related to repeated random experiments, saying that the incidence rate of any particular event will tend towards the probability of that event. Referencing it in this context is a total non-sequitor.
I've used Character AI (https://beta.character.ai) and been seriously impressed with the roleplaying experience that's provided there. A UX feature that really helped was the ability to "swipe left" on a reply and "re-roll" it- this can help keep the AI from getting off topic, or retry when it gives "Eliza" type responses, ie
"AI: Are you ready for your karate practice?
User: Definitely, let's get started!
AI: Great, what sort of things do you think we should do at karate practice?"
If you squint a bit and are willing to provide a little guidance in the form of leading questions, you really can have some pretty fun RP experiences, I've spent hours at this point doing little scenes and I've been really surprised at the wealth of different experiences that the AIs are capable of providing.
Other caveat of course is that it's not really suited to "longform" RP, I can't imagine it scaling to a "campaign" that you return to multiple times per month over the course of a year- I think this is a limitation of the tech at this point, as far as I know the LLM basically is always re-reading the entire chat history to generate the next response and presumably eventually this stops being feasible.
Sigma algebras are useful even in a finite setting where we don't have to worry about pathologies. Think of them as modelling the lack of complete information in a probabilistic setting. If I know exactly which sample point represents my state, then I know the exact value of any random variable. If instead I only know that my state belongs to a given member of my sigma algebra, then I have some information, but not enough to necessarily pinpoint the value of a random variable.
In fact, the familiar tools of measure theory can take this intuition further. If a random variable is measurable with respect to a sigma algebra, then knowing which element of that sigma algebra my state is in actually is sufficient to pinpoint the value of a random variable.
Maybe to make this more concrete:
Let's say I'm going to do two coinflips. My probability space is {HH, HT, TH, TT}. You can check for yourself that the sigma algebra generated by {{HH, HT}}, {TT, TH}} is not the trivial one- this is the sigma algebra that represents "Knowing the value of the first flip, but not the second".
If we let X_first and X_second be 1 or 0 if the first or second flip is H or T respectively, then X_first is measurable with respect to this sigma algebra, but X_second is not.
With Martingales and other stochastic processes, we don't generally have just one sigma algebra, but a sequence of sigma algebras called a "filtration", where each sigma algebra is finer than the last (ie, contains more sets, therefore gives you more measurable random variables). This filtration sort of defines the stochastic process- it's encoding the slow drip of extra information as the stochastic process evolves over time.
I've had some of the biggest belly laughs with a group of friends playing a game of microscope, watching as silly one-offs from the start took a life of their own and mutated over the course of centuries. It's also a really great way to "worldbuild" a universe that you'll then get to play in in a more traditional pen and paper RPG!
In this example, you're not living on the surface of a donut shaped planet, you're living in a 3D space (not a surface!) that is the 3D equivalent of a donut.
Pacman lives on the surface of an actual 2D donut, when he goes to the left side of the screen, he pops out on the right side, and when he goes to the topmost part of the screen, he comes out from the bottom. (Not convinced this is the same as a donut? Imagine the surface was made of a stretchy film and bend the lefthand side to meet the righthand side, forming a cylinder. Now, to make the topmost side meet the bottom side, you fold the cylinder into a donut shape!)
This is the 3D version of the "Pacman universe", where if you go up enough, you come back around the bottom, and the same for all the cardinal directions.
For a sphere, the location of where you land when you go off the screen is a continuous function of where you started from. If two Pac-Men exit the screen next to each other, they will re-enter the screen next to each other.
The real Pac-Man game is a donut because it's discontinuous at the corners. If two Pac-Men are right next to each other near the top-left corner, and one exits via the top and the other exits via the left, they will end up on opposite sides of the map.
There's a mathematical formalization of this, where the thing you look at is closed paths of Pac-Man leaving a point, traveling around, and returning back to that same point. You group such circuits by whether they can be continuously deformed into each other. The discontinuity at the corners makes two distinct families of circuits, which correspond to traveling on a donut around the circumference vs going through the hole.
A donut is a way to embed a two-dimensional torus in our three-dimensional space. What we have here is different. It's a visualisation of a three-dimensional torus. On a two-dimensional donut, there are two directions which loop around. In the space shown here, the only difference is that there are three directions.
A three-dimensional sphere also loops around, but it's not quite the same. One way to get the three-dimensional sphere would be to glue each points at the cube boundary to every other point on the boundary. One way to show that this three-dimensional sphere is not the same as the three-dimensional torus is that in the three-dimensional sphere, you could gather up any tied rope by passing it around the cube boundary.
Normal is actually a stronger claim than "contains any finite string as a substring". That normal numbers contain any finite string as a substring is a straightforward consequence of the infinite monkey theorem: https://en.wikipedia.org/wiki/Infinite_monkey_theorem
I think there is a subtlety here that makes this fail. Normal does not imply that any finite substring exists, just that the probability of such a string existing is uniformly distributed within the space of possible values. There isn't any guarantee that you will actually see such a string, though you almost surely will.
