Summer of Math Exposition

This is a great topic for a video. It’s Very interesting to see a non-intuitive fact about coinflips, which always seem so simple.

I think it could have used a better visualisation. The most intuitive part was definitely the diagram for ten flips. Is there a better way to expand that out further?

Also you could run 20 simulations on the screen at once. That way the viewer could watch as it plays out rather than just see a graph of the results.

Overall it was a good topic and the animations you chose worked well. The link at the end back to real life sports business etc was a great conclusion.

Nice visuals with MS Paint illustrations ! I like your drawing skills, which help to convey emotions.

The topic was nicely motivated with the stairs, although the graphs at 0:40 don’t have any axis labeling or legend, so it wasn’t immediately clear what is depicted (but this became clear shortly afterwards).

Some technical issues unfortunately hindered understanding and decreased clarity:

• You mix a lot of different font faces. Try to be consistent. For some of your fonts, the typography had weird spacing, e.g. consider the word “even” at 3:13. This made these phrases hard to read.
• More importantly, a lot of elements are only shown for a split-second on screen. E.g. the “It is 485 times…” at 3:13, or the “82%” at 4:37 or the final percentage at 4:43. Or things like at 9:04, where in just a few seconds you start a new chapter, show many formulas/text and then give the viewer no time to think this through. More work on pacing is needed to improve clarity.

The script and structuring of how you pose and answer questions throughout the video seems to bear quite a lot of resemblance with LLM-generated responses, which is not bad per se, but just something to keep in mind if you want to find your more personal narration. And this is just my feeling, so take it with a grain of salt.

On a technical side, your audio is very clean without noise. It’s just too bassy and doesn’t have a mono-centered feeling to it (maybe some weird microphone placement?). Also it seems that you’re switching rooms, which makes the audio sound different throughout the video and thus breaks consistency. Some thunder was unexpected at 2:05 and 7:28.

The video is 1080p, which would be nice if elements were also shown in that resolution. This doesn’t seem the case, e.g. at 3:00, the text is blurry and feels upscaled instead of crisp. Resolution of elements seems to change throughout the video.

Pointing arrow animations in red are too aggressive: they repeat their loop all the time and draw your attention to them, even though you already move on.

In general, quite an interesting topic that may need some more thoughts on how new formulas are introduced and explained.

The first half of the video was very entertaining; the drawings were funny and humorous. Halfway through, the pace sped up and I found it hard to follow. Also, the sound quality isn’t the best.

Really awesome video, mindblowing explanation of leads and streaks which I actually didn’t know. Really enjoyed the wojack-style animations, had a good blend of humor and entertainment to keep me watching while also learning the math. I did get a little confused once we started getting heavy into the math with the choose functions, but probably just need to sit with it and stare a bit more. I liked the calculus at the end showing how the integral turned to arcsine and where the name came from. Overall really well done video! One slight audio thing would be to EQ out more of the bass on the vocals to reduce muddiness (I know not a math suggestion, but the music production side of me had to speak up haha). I really enjoyed this one, thanks!!!

The MS art is surprising great, the perspectives are quite attractive.

The topic is really what a lot of people misbelieve, so it’s really useful. And the explanation is clear as well.

But I do think resolution is kind of bad.

Nice presentation, nice topic, creative drawings.

A well put-together lesson probability and random walks. The animations and drawings were memorable and helped to keep engagement and attention once the formulas started getting technical. I did think something was wrong with my speakers for a bit because the audio sounded like it was underwater but once I checked it against some music, I realized it was just the recording. It would be worth borrowing a mic for the next video which I would look forward to seeing.

Overall I like the video. It does a good job describing and illustrating the math and the issues involved.

The style works for what I assume is a young creator. Consider how you can evolve your style as you age and whether the art style would still work for your audience.

One unfortunate element is that near the opening of the video you put me at odds you with by declaring that I would “never trust a 50/50 again” which I just really disagree with! I have similar concerns with the title of the video. Maybe this ends up working out, because I paid attention for errors in your analysis but could only disagree with a couple qualitative conclusions like this (e.g. contrary to your claim at 12:49, I think the winning team is just luckier in your example). If this was a deliberate provocation, consider how to hook the audience while being truthful.

If you disagree with me, consider how you could persuade a skeptical audience on this point. Because while you correctly point out that one or the other is likely to be winning most of the time, it’s still 50/50 on who that is. So I see it as still fair.

But again, overall good discussion of something tangible that most people probably don’t think about.

Nice explanations and subject. I am not a fan of then constant joke visuals that do not bring much content. I am not exactly convinced by the arcsin law argument. Because yes you should look at the proba of being tied, but it should be at any given time, so the sum of proba of being tied and this become a lot larger. Because being tied at any time would indicate a reversal of who is the winner.

The stand out features from this video for me are: the story-telling, the visuals, and the subtle humor. The details in the middle were somewhat difficult to follow and I like the spirit of the examples given at the end of fair processes, but those real-world examples are a little too complicated to be represented by coin flips in my opinion. I do however like the philosophical/life message about how even a fair process can have seemingly unfair outcomes.

This is a really counterintuitive result, and you do a great job building that intuition. I really like the graphics and the humor.

There are a few points where you invoke tools without introduction (generating functions, Stirling’s formula). I know justifying those isn’t the point, but just a brief explanation of what they are would go a long way toward helping viewers follow the steps.

This video, which has interesting content, and good production values commits the cardinal sin of any math explanation. The variables and notation came thick and fast with zero explanation or definition. Slowing down the video and providing explanations would have really helped. I’m sorry because I got the message about the victory (advantage) being sticky even under fair circumstances, I would have liked a better shot at following the math.

The topic is very nice and the hook is intriguing , but some of the formulas just seemed to come out of nowhere , for example $P(L_{2n}) = u_{2k} u_{2n - 2k}$.

Well motivated. The pixel art works because you show the math cleanly. Not sure I understand (or agree) with the way you describe the math. I would have liked more discussion around why you compare a continuous bell curve with the discrete arcsine curve.

your are overloading your microphone, which sometimes makes it hard to understand you, without subtitles.

Really enjoyed this video! The beginning is catchy and makes you want to stay for the full explanation. The sketches are funny too. The sound could be improved a little, but I still prefer this over many professionally made videos with background music, especially when you’re trying to focus on the math.

i love how this points to a simple concept of a coin flip and finds an interesting fact about it that i never thought of and explained it well and supporting math was explained well also

An interesting video. I enjoyed the hand drawn animations, it brought a lot of humour and charm and it isn’t something you see often. The concept is unexpected, and I feel you showed why this unintuitive result actually happens, and explained it well enough, although it was a bit hard to keep up in the middle. Two points I would say is that: 1. I feel that the script sounds slightly AI generated in the way that it flows or the “It’s not X, it’s Y” phrases that are there. 2. The Manim animations weren’t smooth and could’ve really benefitted from a higher frame rate. However I want to say that the basketball example at the end really crystallized the whole thing for me; it was the “aha” moment of the video, because it reminded me of real examples I have seen of this concept. Overall a good job!

I really like the vid! It has a unique sketchy animation style, it’s very charming, and definitely stands out from the rest. The subject matter was pretty interesting, the video does a good job of explaining a counter-intuitive phenomenon in statistics.

However, the formulas/text on the screen go by too quickly to read sometimes, and the audio quality seems a bit inconsistent, I can hear background noises occasionally. Some background music would’ve been nice to have as well.

This video really kept my attention, and all the intuition was nicely explained with grounding in the math as well as the cool illustrations. Super clear. One little thing I noticed was how the manim sections seemed to be lower in resolution/framerate, so probably render settings were a bit off but it didn’t detract from the experience much. Also I would’ve liked to see a bit of working in the generating functions section just to pause and read through

The problem presented is an interesting one, and the applications explained at the end are nice. The intuition in 3:23-5:23 is nicely explained. However, several aspects of the exposition are confusing, and at some points it looks as though you (or possibly an AI that you used to help you?) included parts of the material because you saw them in a textbook without clearly understanding how they relate to the rest of what you presented in the video.

• It’s not remotely obvious to me how the explanation of the reflection principle explained in 5:24-6:14 is meant to relate to the problem at hand. You say that this is how mathematicians have thought about the problem and that thinking about the problem this way is really strong for building our intuition - but you have not explained how this is the case.

• The explanation in 8:40-8:56 was not at all clear. The explanation at 8:57-9:17 looks like it’s intended to assume advanced university knowledge of probability theory, which seems to contradict “high-school” being listed among the intended audience.

• The presentation at 0:15-0:37 is misleading, because it makes it sound as if the following is the case: the “1,200,000” was simply a large time at which we decide to compare steps, and the fact that the two step counts at this time ended up being similar is reasonably to be expected because of the fairness of the coin. Of course, that is not the case: if we arbitrarily pick some large time and decide to compare steps, the likelihood that the step counts will be very similar is pretty much zero. Instead, the fairness of the coin merely has the consequence that there exist arbitrarily large times at which the step counts are close to each other, and the hypothetical scenario presented is simply one in which one of these times happens to be about 1,200,000.

• In 2:28-2:39, the concept of a “random walk” is introduced in an incoherent manner. What you mean to say is that a sequence of coin tosses can be used to generate what statisticians call a ‘random walk’, by defining a motion in which each Heads means moving up one unit and each tails means moving down one unit.

• It took me quite a long time to work out what’s going on at 2:40-3:01. One of the several sources of confusion was the fact that the text under the middle stack incorrectly says “two up, two down” (which would mean two heads and two tails, as per 2:26-2:37) instead of saying something like “two ahead, two behind”. The other point that hadn’t been made clear was that what you were counting was the number of joining line segments that lay above or below zero. (I think this same clarification might also have been helpful at 8:03-8:08 when describing Feller’s result.)

• In 1:40-2:15, for the axis labelled “frequency”, it’s not clear what the set of trials is whose frequencies of results are being counted.

• Did you ever explicitly explain the link between the random walk and the stair-climbing match (namely: the signed difference between you and Jimmy in the stair-climbing match is a random walk, and hence, positive and negative for the random walk correspond respectively to ‘ahead’ and ‘behind’ for the stair-climbing match)? I myself understood that this is the intended link, because I have experience with these kinds of problems; but for someone without such experience, this is not obvious and should be made more explicit.

Finally, I think it would have been helpful to provide some reference so that those who want to read up more about the presented results can do so. (For example, just searching the phrase “Feller’s theorem” on Google doesn’t seem to readily return the particular result that you’re talking about at 8:02ff.)

I liked the art style and the video was overall quite informative. I believe, that at least to an audience that has prior knowledge of statistics, the seeming ‘paradox’ of the arcsine law is not that paradoxical. For that reason, I believe focusing more on the mystery curve could be a more interesting video, such as diving deeper into the Feller’s theorem (although I am not sure how complicated that gets); although, I do understand that this video has a lower barrier to entry than the one I described.

I think the concept and motivation behind explaining the problem was quite cute. Most of the explanations were intuitive and easy to follow. Nearer the end when deriving the arcsine law, it starts to get a bit more trickier to understand, but it’s easy to visualize the general idea behind why this is the case and why this would be relevant in real life.

This was a great video. It takes a common experience, a fair coin flip and uses it to introduce some sophisticated ideas - random walks, Markov chains, the arcsine law - and ultimately draws an unintuitive conclusion. The plots showing the random walk over time was an excellent visual aid. The math was rich and relatively easy to follow, however I don’t think reading equations adds much value - “u sub 2k times u sub 2n minus 2k” - talk about what they are, the audience will read the equations. I was particularly interested in the reflection principle, but I could not quite see how the subsequent combinatorics calculations followed from there. I enjoyed the narrative, it was relatable and provided strong motivation for the explanation.

Some interesting weather in your background, I think :)

Loved it!

I liked this video a lot. The animation was pretty good. There are a few critiques but overall not many and the critiques I had were mentioned in someone else’s feedback as I had originally watched it with them so repeating it wouldnt really be necessary as a whole though I thought it was a pretty good video

Great video and well-motivated! The painting is awesome, haha. I really enjoyed it.

Pretty cool, learned something new

Some of the pacing was off - things were shown too quickly and probably could have been cut

Parts seemed too AI-esque

The discussion at the end about the “real world” implications went on for a bit too long

Would have been nice to dig into the derivation more. why is there pi involved?

Interesting material and presentation. It would have been more effective if you established a clear picture of the target viewer and their level of knowledge about probability and statistics, and then built from that level towards the more advanced concepts. The video had a mixture of very elementary concepts mixed with jumps to more advanced concepts. The hand-drawn illustrations were an interesting break from precision mathematical graphics but may have detracted from the mathematical exposition.