Summer of Math Exposition

Great video!

The topic is interesting, the progression is logical, and the animation makes everything clear.

Some minor nitpicks: Add short pauses between your sentences when you switch problems to signify a change of topics. As it is, they run together a bit, and a viewer who needs to gather their thoughts doesn’t have a natural stopping point. When you introduce the optional stopping theorem, you present it as an interesting aside, not a useful tool that we’ll need going forward.

The topic is very good, the visuals are really nice, but one problem is that it feels a bit repetitive, doing similar calculations over and over again. Also, surprised that you didn’t formally define Markov chains or matrices.

This is an interesting video, and the creator makes good use of video as a medium. I was not familiar with this topic, and I found it relatively easy to follow and fun to watch.

One could argue that the video moves just a touch too fast. For example, I had to stop the video once during problem 3 to understand why the probability of 1/2 was multiplied by 0 in going from HH to HHH. It would have been nice for the creator to emphasize once again that “c” represented the probability that Alice wins starting from the point HH. There were a few other points in the video when something was stated quickly that wasn’t obvious (to me).

Another minor criticism I have for the creator is that there is a whole lot of, “so we end up with this equation.” I find such statements a bit “tedious” and off-putting. it would be good to think about different ways of saying this so that you are not always saying the same thing over and over again.

What worked well & What I especially liked:

This was by far the best video I have watched! The progression from simple to complex problems was exceptionally well-structured, the pacing was spot-on, and the animations were top-tier. Using Markov chains and state diagrams to make non-intuitive probability problems feel systematic is a brilliant topic choice.

What could be improved & Actionable suggestion:

The video is already amazing, but there is one small visual step regarding the very first diagram presented in the video that could make the initial concept even more intuitive:

Instead of jumping straight to the self-loop in that first diagram, it would be extremely powerful to start by unfolding the process into an infinite tree of states:

Show the initial Start node branching into two outcomes: either moving to the Stop state or moving to an identical new Start node.

Repeat this branching step a few times to visually illustrate an infinite chain of identical sub-trees.

Highlight the self-similar pattern in this infinite tree to explain why we can fold it into a single state with a self-loop (Markov property) and set up the recursive equation a = 1 + (1/6)*0 + (5/6)*a.

Showing this transition from an infinite decision tree to that first compact state diagram would make the recursive nature of the equation crystal clear to anyone seeing state diagrams for the first time.

Overall, an outstanding, super engaging, and beautifully animated submission!

A good collection of problems which incremented in complexity very nicely. The problems were worked cleanly and thoroughly. It was interesting to see these computations without matrices or explicit Markov-chains. The video was harder to follow or less memorable in a big picture perspective. I had forgotten the problem at the beginning of the video by the time it came up. I remembered that there was an overarching problem we wanted to solve, but I didn’t realize we were actually tackling it until after the video was over because there was no callback when the problem came up again. Would rate it even higher if there was more flow between problems and referring back to previous/future problems. I also really liked the historical contexts for the problems presented.

Very nice video. The motivation is great: these are fun example problems you’ve got, and I think we’ve all had the experience of trying to do a probability calculation but not really knowing how to start because the question doesn’t feel amenable to basic probability rules, and these tools really help with that!

I do think there are some minor presentation issues. The bigger one is that the exposition has a few weak points. For instance, the logic behind the equations you write sometimes goes too fast, and actually I don’t think you stated clearly at first where they come from. With the last example as well, the logic goes a bit too fast or is unclear, with new principles being introduced without much acknowledgement. Finally, there are a handful of moments when the pauses between sentences seem a bit too short, mostly near the beginning of the video, presumably an editing issue.

But overall, I think this is very good work, and jams a lot of value into a short time by introducing such powerful yet simple and elegant tools. Nice!

Solid video, well motivated and clear, and explains a very central topic in probability theory.

However, the content is very textbook with minimal innovation. It would be more interesting to have some unique insights or advances in pedagogy beyond animating the state diagrams.

Amazing video, beautifully explained. Some of the explanations behind equations (e.g. 1:26-1:48; 8:13-8:16, 8:33-8:36; 16:32-17:03) were quite fast, such that perhaps not many people would follow them first time round, and a high-schooler would likely need someone else to explain what’s going on. Still, I can see this video being an excellent and inspiring resource for a teacher or lecturer to work through with students.

The best video I have seen in SoME5 so far. Beautiful solutions, explanations and visualizations. Nice choice of problems as well

I was confused with all the numbers, and I don’t even feel an intro to Markov chains in matrix terms is more diffucult, quite the opposite

Good video, some comments: references to additional reading or other videos is a great way to help people know where to go next. Having some example problems would be a great addition to this video, you mentioned to try them in the video, but most people won’t, but if you present so interesting problems without giving the answer right away, some people will bite. (Maybe explain them in a different video or blog post?) Thanks for your contributions.

Hello! Thank you for this video, I feel lucky that I came across it. I think it has tremendous educational value, a great and clearly experienced method of teaching by breaking complex problems into elementary steps, and it provides the viewer with a method that will help them do the same for most probability problems they come across, and I’m certainly left with a new favorite way to solve them.

With all due respect, I do have some advice on the video-making aspect of things. Namely, I think the introduction of the video along with the thumbnail/title sell the video very much short:

The video starts with simple problems, going up in complexity step by step to build the concept up as it should. This approach however requires a stronger intro, because a viewer shouldn’t have to take it in faith that these steps are going somewhere. They should already be very assured of that, and should be interested in the how part.

I’m guessing that the goal in adding previewed problem in the intro is to give this message. But that problem also looks very simple, is very specific, and its depth might be hard to appreciate for anyone not already familiar with such problems, including myself.

Personally, I would have been more willing and excited to watch the video, had you teased the method given in the video differently. If I may, here’s a list of pointers of what that would look like for me:

• Make sure you get the viewer to understand that the method you’re offering to teach in the video can be applied to most probability problems (or all discrete ones, if I understand it right) to break them down into these trivial algebraic steps, as that’s the primary appeal

• Use how in the video the method has been applied to historical problems put forth by famous mathematicians - as that shows that the method applies to “important” problems.

• In the thumbnail or again the intro, perhaps tease the connection to the golden ratio at the end, as that is an unexpected “aha” moment that can be very appealing to learn about.

• Show more of the problems solved and diagrams drawn in the video in the intro, to show that this is going to be a journey through different problems in probability, and we will be making a connection between them in how they can all be similarly solved. This shows that you’ve put a lot into the video - there’s variety, it’s not just a 20 minute walkthrough of a single problem. It’s good to have a high “density” of things happening in the intro, that means that the viewer will get a lot from watching it.

I hope that you’ll find these comments helpful, I’d love to see you make more content that will resonate better with the YouTube format!

A few nitpicks to end on:

• Some of the transitions happen a bit too suddenly, for example the transition to de Moivre and back on problem 2

• About the same part, in my opinion the video flow is better when you just pop a little de Moivre on the screen, say that the problem was posed by him without even changing scenes. When the screen is fully switched and you use a phrase like “back to the problem”, it can cause one to feel that there was a digression from the topic, when it was only a fact inserted in.

Thank you, I’m looking forward to seeing more of your work in the future!

I liked this video a lot. I’ve seen state diagrams a lot before, but I’ve often failed to see how they might be useful in actually solving a probability problem. This helped me see they’re use, and I’m definitely a fan now!

I liked the visuals a lot too. They were a big help in understanding everything. Very well balanced with the rest of the video (weren’t too much or too little)

I think my biggest complaint is that where you got the algebra from didn’t always make sense. Additionally, the pacing of the voice over is off. I feel although it’s very constant and doesn’t leave a lot of room for me to think through with my own thoughts. I assume it’s based on a script; it’s just lacking the normal (very brief) pauses of someone explaining something and kind of thinking through it on their own. If the voice over was done a bit more conversationally or naturally I think the video would be super good!

The natural evolution of problems was also great. I think the slow progression was well done.

Super clean animation style and nice introduction to the topic. A real-world application would have greatly helped solidify the concepts. These all seem like toy problems which might be interesting in-and-of themselves, they’re all coin flipping. What about probability of wininning BlackJack, winning a lotto, etc.?

Very interesting topic. Clear animations and explanations. Well done! My only suggestion is to expand the explanation more when you discuss how the state probabilities are calculated.

Exceptional. I enjoyed this a great deal and felt like I learned something important. As an engineer who hates statistics, this takes the exact kind of problems I avoid and expresses them clearly, in a way that I enjoyed watching the solution. Your tone is clear and approachable. The economy of this video is impressive as well, not a wasted moment or frame. I find myself asking if there are places in my professional life I could apply this. I like how you built up from simple problems to more complicated problems to problems of infinite size. Just a wonderful exercise in compound learning. Keep making videos, you have a new subscriber.

Nice animation. Material not particularly challenging. Very happy sounding voice.

Might have been clearer if you gave the probabilities for the stop states to motivate a general formula for the equations for each state.
Some repetition in the script feels unnatural/scripted, e.g. “this is the famous []”.

So first of all, you have a great and highly unique animation style which I absolutely adore. Also your explanations are all in all very clear and you build up the concepts smoothly through increasingly complex examples without any unnecessary formalisms. The reason I didn’t give you a Score of 9 is the interplay of the following minor points:

First of all it maybe would have helped to explain why we add 5/6 * a and not something like 5/6 * (a-1), so why we indeed need “a” tries again if the first one failed (i.e., explicitly talking about the Markov property/memorylessness).

Also you could have stated already at the beginning that the variables written beneath the states represent the expected number of remaining tries to reach the goal.

In the example around 7:30 explaining the reasoning behind the state transitions could be slightly more explicit. You could have pointed out how rolling HTT is “useless” (it doesn’t overlap with the start of either sequence, resetting us to the starting state) whereas HHT contains the start for HTH and so we only need an H to get to Alice wins which is why we can transition to state HT.

In the coin flipping example, emphasizing memorylessness again (the system doesn’t know how many times which coin side already appeared) would clarify why a single step to the left always has a probability of a, regardless of where you are on the chain.

As a general thought, it might be nice to briefly touch upon the formal definition of Markov chains at the end, after building up the intuition for students who struggle with Markov chains and who need the topic for school/university. However, if your goal was purely to explain the core concept without any formal theory, the video already works wonderfully as is.

And of course, these are just my thoughts on what would be useful for someone with zero prior knowledge about the topic.

Anyways, definitely keep up the good work!

Neat and nicely explained!

I will say, though, it does feel a bit strange that you frame the video as about solving probability problems with state diagrams, but the solution to the final problem required more than just state diagrams to solve. Still, it was cool to see how you resolved the final ambiguity in the end!

Extremely well-structured and highly instructive. Regarding the final example, I feel the derivation of b = a² moves a bit too quickly. It might have been worth highlighting the figure’s symmetry (though, admittedly, re-watching the segment allows one to work out the argument independently, which has its own pedagogical value). One minor quibble: the video could have concluded by opening up a broader perspective—perhaps touching on more general problems, such as those involving continuous time.

I think the idea of using state diagrams to solve problems in probability is actually a good concept to illustrate how to do so. I think more emphasis can be explained on why certain probabilities or expectations do decompose the way they do, as the explanations are a bit fast, and it is a lot of algebra on auto-pilot. Slowing down would help the audience help to grasp why the decomposition works . I think there’s also a few assumptions missing here. (including independence only conditional on last state).