Summer of Math Exposition

It’s perfect for the most part. The only thing I would love to have here is division of the video into different chapters.

Very good

A great video all around. However I disagree that this is a good “math explainer” for someone just getting introduced to probability (even at the undergraduate level), as the interpretation of “the 1,000,000th digit of pie being even or odd” having two possible answers depending on the Bayseian or Frequentist lens might be confusing for those just getting into it.

The motivation (intro) and memorability (conclusion) are great and intriguing for someone who already has a solid grasp on the topic. As well as the intermediate chapters explaining set theory and probability were well done. I just don’t know if they would go together as a math explainer.

If I personally were to show this video to curious high-school students (who would be new to the idea of probability), I would probably leave out the into and conclusion.

The video is clear and has some nice graphics that should help to keep the viewer engaged. However, there is an awful lot of words, and they go by fairly quickly, so unless the viewer is particularly interested in this specific topic, they might end up getting bored. That’s kind of how I felt. I’m not particularly interested in probability, but I am somewhat familiar with the topic and am open to learning more. Unfortunately, I found my mind wandering as the video went on. Again, the information was clearly presented, but it felt a bit like reading a textbook and didn’t really capture my attention the way a great video might.

I learned some valuable things through this video. Great animation, good audio and speech flow. Ideas were presented well with great visuals. I genuinely learned some new thing. Thank you.

“Thomas Interpretation” was an odd moment, lol. Immediately went to read the comments about it on Youtube.

Very good introduction. Starts off with a nice hook aobut the digits of pi. From there on, nicely paced, well animated etc. nothing really to complain. Not the most original topic if I may point out somehting. And perhaps the music is just a bit loud.

This is a really good introduction to probability. The content itself is nothing groundbreaking, but the ideas are motivated, the progression is natural, and the visuals build the intuition. Well done.

While the premise of the video was not necessarily novel, the video itself was very clear.

Great presentation of basic probability, the two interpretations will stick with me.

I wish we could have explored the belief interpretation more deeply and/or expanded on the ‘masses’ concept using examples involving infinite sets, since these new things will confuse people most and will help them appreciate the need for this rigorous treatment of probability that the video builds upon, but it is understandable why it wasn’t done.

I learned a lot of the theoretical definition behind probability. This was frustrating in some ways because I was hoping to learn how to figure basic probabilities. The idea of probability covering confidence rather than mere chance was new to me. I did not like a lot of the visuals which were filled with scary looking math notation. It is possible to describe sets with pictures and the trembling or sweaty hand that flips the coin without calculus notation. And unfortunately, knowing the theory of probability does not feel nearly as useful as knowing how to figure a simple probability that explains why coincidence is possible or how a game works.

Really nice

I think this is a good video. It explains probability really well. My only qualm is that it might be difficult for some people to keep up, and a few parts could be a bit confusing. Apart from that, very good video.

Just a very solid intro to probability theory.

I really like this video. It’s a good introduction to probability.

I just have a few notes:

• I quite like the dedication to accuracy. The fact that at 0:21, you actually have the correct digits for the 999,995th - 1,000,018th digits of pi is commendable.

• I would recommend showing the footnote at 8:33 for a tiny bit longer. It’s hard to pause at the right time.

• At 10:39, while it is likely beyond the scope of this video, I wouldn’t say that events with probability 0 are impossible, since that’s not the case with uncountably infinite sample spaces. I know you are focusing on sample spaces that are countably infinite, but a footnote there would have been nice.

• At around the 12 minute mark, I believe you have the Kolmogorov axioms wrong. One of the axioms needs to be that probabilities are nonnegative. And your axiom 2 is not really an axiom. From $P(S)=1$ you can derive $P(\emptyset)=0$ by considering the intersection. We know that $P(S\cup\emptyset)=P(S)+P(\emptyset)$ since the two events are disjoint. Since $P(S) = 1$ and $P(S\cup\emptyset)=1$, that forces $P(\emptyset)=0$.

• At 17:30, I quite like the joke of Thomas Interpretation. Probably a better mathematician than Euler.

• And I want to say you did a really good job of explaining the difference between the two main interpretations of probability, even if the Bayesians are objectively wrong. /s This is especially true as you gave examples for when to use each interpretation.

• I would have liked to see the solutions to the problems at 21:10. They were a fun exercise to try to do myself, and having the solutions in your video description or a comment would have been nice to check my work.

Overall though, this is a really good video. I really enjoyed it.

Good framing around the problem of things appearing deterministic, but practically being probabilistic

Personally I didn’t like the hook but after watching the video I get why you chose it. You effortlessly introduce the difference between deterministic and stochastic processes.

your animations make clarity top notch. All movement is motivated and lines up with narration. Big big win. The quick set theory explainer helps too. The language does use jargon but not without explaining it first.

At 1:20 you try to overwhelm the user with equations, it does work but the math starts to feel dense and nonsensical like it is random. This is an odd feeling, if you’d want to use that maybe awe is a better emotion to aspire to. The examples that you give (weather) have that feeling already sprinkled in since most of the systems you mentioned are verifiably chaotic and therefore stochastic in their nature. That means deterministic formulas with stochastic outcomes, in the grand scheme of things, due to sensitivity to initial conditions. Rather than telling I would SHOW the user, that gives the feeling but saves you the narration time.

The subject isn’t new but you are one of the few that has a great headstart for explaining the frequentist vs bayesian framework. You already do this and I am hoping you’ll expand some more on it.

In terms of memorability. There are definitely good concepts that will stick with me. But your entry has one fatal flaw and that is the design. Language, visuals (manim) and sound (same soundtrack) are all identical to 3b1b. I understand that this is an established brand/method that is finetuned till perfection, so diverging from that method usually means your video becomes worse rather than better.

But for me and other users to associate this video with you, since you are putting in the work here, you will have to create your own identity/design. I won’t lower your score for it, but I found it important to mention.

so good

It’s a good intro but it seems textbook

This doesn’t offer new insights

The visualizations are clear

Excellent video. The explanation of probability was really interesting and efficient. The voice is clear and the music helps setting the mood. The Manim animations are well-thought and nicely presented.

The only criticism would be to try and find something that differs a bit different from a 3B1B video, as this is really similar in every aspect, and deserves to have its own style (at least in some parts), rather than living in the shadow of 3B1B

Excellent work.

Your deadpan delivery of “Thomas Interpretation” is great. Very undersold joke.

I think the first 15 minutes of this video were very strong and would indeed give a good foundation. I could seeing it being a useful follow-up to a day 1 / week 1 probability class. I think the final chapter after 15’ fell off for me for a few reasons:

1. The examples of frequentist and Bayesian probability were not very comparable. The tests-positive patient is a classic set-up for analyzing conditional probabilities with Bayes rule, but doesn’t actually get to the guts of the distinction between Bayesian and frequentist approaches. (The covered die roll is briefly alluded to, but a better comparison in my opinion.)
2. The scales of justice graphic between the two camps I don’t think is accurate. There are people who argue only one is acceptable. Certainly contemporary stats has become more ecumenical, but die hards remains. And there are definitely reasons that go strictly one direction or the other, even if overall the decision is uncertain. Perhaps this discourse is better left to future video, or the practical trade-offs could be briefly listed (eg, computational costs, interpretation, integration of existing evidence, etc.)

A nice provocative introductory slide that illustrates a nuance of the philosophy that despite all the years I’ve worked with probability, I still somehow never seriously thought about. I’m still learning new weird stuff about probability every day. Though, the question of whether it’s 50% or it “must be either 0% or 100%” is more reminiscent of some comparison of frequentist vs. Bayesian where population mean as a parameter in one is considered not a random variable, and in the other, it is. My intuitive understanding of probability did make me more or less immediately jump toward the 50% formulation, since “quantifying uncertainty” and “conditioning” are very familiar concepts to me. I am definitely more familiar with frequentist statistics and often think in that way, but nevertheless I rarely ever say something like “it either must be 0% or 100%” … I save that for conditional probability. I know this is supposed to be a super-introductory video, but maybe some mention in passing, or how it means “refinement of beliefs” for Bayesians, would have improved it a smidgen. That said. Great video! Or, it’s the suggestion of what to do next.

Also liked the Thomas Interpretation joke. I bet someone complained you made a typo, though. hahah. (Are they still using that book Conceptual Physics these days … it’s the first book in which I saw jokes of the type).

Excellent video. The video started with a question I did not know the answer to and by the end I understood the answer was “it depends” and understood why it was that answer. It would have been nice to see a more rigorous frequentist interpretation of the 100th digit probability by looking at the numbers of odds and evens up to the 99th digit and predicting the 100th based on which way those numbers leaned up to that point, although this suggestion isn’t necessary for the presentation of the overall concept.

I liked that there were correct subtitles. It was well animated and the explanations were intuitive. Well done!

This was really good - it served successfully as a good intro to probability as well as one of the best primers on set theory I’ve seen. I really liked it. “Thomas interpretation” made me LOL.

Definitely the best SoME I’ve seen this year. I feel like this could’ve been posted by 3b1b. I do think the pacing was a bit slow, but maybe that’s because I already knew the material. Otherwise, it was excellent.

Really good explanation of the basics!

What worked well & what I especially liked: The opening hook regarding the 1,000,000th digit of pi was fantastic! Demonstrating how different definitions of probability (frequentist vs. Bayesian) lead to completely different answers (0% or 100% vs. 50%) for a deterministic fact was highly engaging and memorable. The animations were top-notch and visually clear throughout the entire video, making abstract concepts intuitive and pleasant to follow.

Actionable Improvement / What could be expanded: When introducing the idea of probability as “mass” inside a universe S, the geometric intuition works well, but it could be formalized slightly better without losing accessibility:

• Conditional Context: It would be clearer to emphasize that every probability measure is implicitly conditioned on the sample space S. Rather than talking about “the probability of event A” in isolation, framing it as P(A|S) highlights that probabilities are always relative to the universe of outcomes being considered.
• Set Cardinality/Measure: Explicitly mentioning that in uniform discrete spaces this “mass ratio” boils down to the ratio of cardinalities (|A| / |S|) would bridge the visual intuition directly with basic set theory math.

Clarity, Pacing, & Target Audience: The video succeeds very well as a “Chapter 0” for high school and early undergraduate students. The pacing feels right, and the visuals strongly support the narrative. With a slight tweak to how probability spaces are conditioned, it would provide an even stronger foundation for students entering their first formal course in probability or statistics.

Cool introduction to probability. I like the focus on how interpretation of the probability is inherently decided based on what aspect of the outcome of the experiment we want to know.

There are many videos on these introductory topics but this video does have its unique flair to it.

I think this was a very good introduction to probability that I think many courses out there are missing. I especially like the thinking of probability as a model for randomness instead of the usual frequentist approach, which segue into the Bayesian and frequentist interpretation nicely and discussions of what philosophically we interpret probability as. The example with nth digit of pi could be a bit more concise with the definition so as to not throw misinterpretation here, but overall, a very good video.

Truly a video to match “What Is Random?” by Vsauce. I can’t say it’s super novel because I’ve seen that video, but it’s still good.