How much counting to include in EoP?

Awesome

Kravenar Games

Not only because I would love to see an "Essence of Counting" miniseries.

Jason Orendorff

Thought experiment. Suppose right after showing elementary n/N probabilities, you said, "more complicated situations can be modeled using larger numbers of possibilities. for example, if we were rolling two dice instead of one... In this way, some complicated probability problems are really about counting the possibilities. My 'Essence of Counting' miniseries shows much more..." Then you set combinatorics aside and turn to more statistics-y stuff. Would the viewer be missing out on anything? I don't think combinatorics-heavy probability problems teach much about probability, so I would tend to say *no*, you should actually do this.

Jason Orendorff

Speaking as a layman with a renewed interest in mathematics I would definitely appreciate a primer on counting methods.

I like the idea of a distinct video on counting methods, which could be referenced/cross-referenced when other subjects need counting methods review

I'd vote on saying your audience is pretty versed in at least the basics of most probability/stats, this would give a slight barrier to entry to the casual observer, but there are good resources online and on youtube on these more basic subjects. Either way it'll be great!

It might be a good idea to just explain the basics in the video with a footnotes video if they want more information.

I prefer a dedicated video. It is more reusable by others.

Brian Matthews

It could work either way--almost all books on probability do the counting stuff first, but you were never one to stick to the textbook way of explaining things. Besides, if people want that, it's the thing that is already out there, and common. For that matter, it's even possible to explain probability and never go through the details on how to compute the binomial probability distribution. It all depends on what material you want to get to.

Kevin Iga

I think it's pretty important to do a video on "counting". Mostly because it *is* kind of an important foundation for probability, and with your unique style you could make it very intuitive. More importantly, combinatorics is one field in which most students (or at least me) get very "unintuitive" answers. One of the reasons people think of it as a toy, IMO, is that often, you can do something that feels right, and then realize you forgot something, so your answer is completely wrong now. It's often just a situation of "oh yeah I forgot to divide by this thing" or something. This makes it hard, but makes a good presetnation of it all the more valuable. And btw, one of my greatest "a-ha" moments in learning probability came from understanding where distributions "come from". E.g. understanding their PDF's/PMF's and where the formula came from. And this is hard to do without the counting.

Edan Maor

Exactly right, I assume that a lot of the audience already "knows" a lot of the material from your videos, but likes having another way of thinking about it.

Edan Maor

imo, combinatorics is interesting enough a topic in itself to be treated in a full video. it's not just the choose-function, you can also treat questions like the Mississippi-question, or the bagel-question (<a href="https://www.youtube.com/watch?v=UTCScjoPymA)," rel="nofollow noopener" target="_blank">https://www.youtube.com/watch?v=UTCScjoPymA),</a> or the probability of holding a royal flush in poker

Andreas Blatter

Nice

Illuminati Games

I believe (or more precisely, I have been told that way and I feel that it is the right way) that the basic methods of counting must come prior to any other course on probabilities. By technical necessity first, because the simplest probabilities roughly consist in counting, and maybe by spirit, since you probably happen to discover binomial coefficients before normal distribution.

Thomas Boyer

I love combinatorics problems so I absolutely would love as much counting as possible that ends up in the series. On another note, its always nice in my opinion to dip into other subject areas because for math students it can sometimes light a spark and show them an area of math they love that they may not have found otherwise. It's always cool to explore math on your own.

I'd really like a video on counting methods and I think it could fit in well with the series, be it as an early episode or as a side dish. I am a maths teacher and I used your calculus series this year both as a way to get ideas on how to present the ideas and by directly showing the kids the videos (their summer homework is to watch the rest!). Luckily enough for me, next year we are doing... probability!

Javier Almeida

A refresher on combinatorics would be very useful!

Even as a toy, I find that most finite math books or basic probability books that introduce combinatorics make it look like a *boring* toy, which is even worse in my opinion. Beginning a probability series with n-choose-k sounds dreadfully textbook. But I would not feel offended at all to hear the fib "how do mathematicians count these huge finite sample spaces? With a little toy called combinatorialists", if then instead of a discussion on binomial coefficients (which in my experience most probability students never find enlightening, let alone interesting), there is a link to the circle-cutting graph theory video Grant did a while back, because that demonstrates the exciting structures, deep theory, and creative problem solving technique that using combinatorics entails.

Sean Bibby

Honestly, maybe link them to your circles-cut-by-chords counting video rather than a dry and ultimately inconsequential explanation of n-choose-k, because that is way more enlightening as an explanation of "what do combinatorialists do to count these big sets".

Sean Bibby

Saying this as a combinatorics guy: just explain it if it ever becomes necessary, maybe in a side video, but I doubt it will. Really combinatorics is a way to not have to "count them one by one" (I drew all the posets with five elements once), but to actually start with probability, you can just say "let's say we listed all the possible outcomes and there were 2,047, that's our sample space", and go from there, because how you got that number is not important for probability concepts. Then maybe all that is truly necessary for understanding is a throwaway comment: "hey, maybe you're wondering how mathematicians manage to not spend weeks at a time counting just to do the one simple probability question? There's a field of math called combinatorics that investigate time-saving tricks and concepts for understanding and counting large sets!"

Sean Bibby

Agreed, showing that limit at some point feels like a must. Likewise with the Poisson distribution.

3blue1brown

Great points, I think it's definitely right to motivate various counting techniques with clear probability problems. Also, you are probably right to call me out on the real-worldness claim, I suppose what I mean is that the examples one tends to see as combinatorics is introduced have a tendency to be to models.

3blue1brown

The option of dedicating a primer video to combinatorics appears much better to me. Those who know about the subject (or believe, or wonder if they do) will be interested, as well as those that don't. But mainly this allows you to refer to that video whenever needed, in other videos. I also guess dedicating that primer video to the subject might (I hope) give you the opportunity to add other additional videos to the subject.

Start with the basics. These videos aren't just for your loyal patrons. They are--and should--be a resource the whole internet can use in perpetuity. I don't mind a refresher on combinatorics. Even if I learn nothing, your animations and soothing narration are always enjoyable in their own right.

jason black

First of all, even if it's standard content, it's beautiful and you'll give a unique flavor that hasn't been achieved. For example, what if, later in the course, you added an appendix video demonstrating how a normal distribution can arise naturally as a limit of binomial distributions? Use a model of people's height as an example: say there are 100 or 1000 genes that regulate a person's height which can be "on" or "off", and a person's height linearly depends on the number of "on"s they have? With your animations turning bar graphs into smooth regions, this could work really well for explaining how a continuous phenomenon - a normal distribution - comes from a discrete phenomenon - a binomial distribution. If you want to expound that theme, this seems like one of the examples you should want.

Jacob Mirra

Essence of Probability, I assume.

Elliott

I think your style could make for an interesting enough cover of counting to warrant its own video. You could release it simultaneously and make it optional so the other topics aren't slowed down by long asides. But as someone who does statistical analysis every day, I'd really appreciate a visual refresher on the topic.

Elliott

What is EoP ????

Michael McGuffin

Counting (or combinatorics, as it is properly known) is important, in its own right. A lot of problems in computer science break down to counting things. For example, if you want to know how fast an algorithm is (using Big O notation), you need to have some idea of how many elements of the input data structure you will need to traverse. The entire field of complexity theory is arguably just a special case of combinatorics, and probability never even enters the picture. If you treat combinatorics as a "toy," you may cause people to believe that it has no applications outside probability theory, which is problematic for those of us who actually do things in this non-probability space.

Kevin

I think that for sure you need an aside or even a supplementary video to explain the choose function and combinatorics, I get the feeling that the first half of these videos is going to involve a lot of discrete probability problems, before moving into continuous later and the choose function is essential to understanding the early stuff like counting probability. Sure, most of us will know choose and Pascal like the back of our hands but I feel that you the purpose your videos have always had is to inform people intimidated by their first path at math and I think it'd be a huge mistake to ignore that audience

Maybe do it as one of your regular videos which is not part of any series. Counting in itself can be pretty interesting/challenging so I think a lot of people might benefit. Plus without having it tied into this series you can go a little further into advanced counting strategies.

The people who watch these videos may already know the counting stuff, but if they're like me, they enjoy the pedagogical nuance you bring, which at worst is nothing new but at best is something fresher. In this regard, if I am right, I think you can satisfy the majority of all audiences. I.e. explain everything using the techniques you'd like, in assurance it'll satisfy the whole audience. Even if it's on different levels. I.e. The way a great "kids" movie works for adults and kids alike.

Matthew O'Connor

In my experience pascals triangle can be seen from two different perspectives. 1. You can see each point in the triangle as having a number, which is the number of ways (number of paths) you can get to that point, by moving down from the top point of the triangle (This seems synonymous with the binomial coefficients view to me). 2. You can think of each row as being about a single decision, for example "will I eat a 4 donuts or a salad for lunch". Each point in the row will then randomly decide with a 50/50 probability which choice to make. The distribution down at the very bottom could then be peoples weight.

Myles Buckley

I think a primer video on combinatorics is a good thing as well. I took a class on it and enjoyed learning about the different counting principles. I felt it was very helpful when I later took the probability and statistics course then. Maybe you could tie it to the cryptography you were doing. For example, counting the number of possibilities in trying all the binary combinations. You gave some numbers to it but maybe use combinatorics to show how a computer or set of computers would go about randomly solving this. Show how combinatorics would be used in calculating this.

I too like your plan with a primer on counting, then one on what distributions and after that getting into binomial and poisson. I'd like one on combining distributions later on too.

Peter Bork

I think it's a little weird that so much emphasis is put on the binomial distribution, and different classes of probability distributions in general. These ideas seem quite separate from the bayesian theory. My experience is that empirical and normal probability distributions are much more common than these special classes in real-world applications.

Julian

This is good to hear. I'm starting to agree with others here that a primer video would be worth it, but you bring up a good point that it's important to clarify what role these examples/techniques have for more the more general and useful concepts that follow later on.

3blue1brown

Great perspective, thanks!

3blue1brown

I agree with Kenneth Goodman that a primer video that gives a solid introduction to factorials, chooses, nPk, etc. it extremely important and would be a useful reference to anyone studying probability, computer science, discrete math, etc. Breaking the mystery of these formulas should happen early, and reminders of what they mean as they come up in future videos should happen briefly but often. As a lot of facts in probability can be quite mysterious and counter-intuitive (Monty Hall problem for example), clarity on definitions and basics I think is a must. Just my two cents!

Hey, two thoughts: (a) whilst it may take more work, why not think about showing some combinatorics *through* probability? Meaning using probability questions (decks of cards if you want something cliche) to motivate learning combinatorics. The first chapter of Casella &amp; Berger (<a href="https://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126)" rel="nofollow noopener" target="_blank">https://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126)</a> has some good examples. I envision something like you filling the 2 x 2 arrangement "table" (sampling {with/without replacement}{order matters/doesn't} using four probability inspired examples. (b) I would strongly disagree that "combinatorics tends to be more relevant to toy problems than to real-world modeling". Many things in Statistics need the very basics of combinatorics to construct/correct likelihoods, for instance. Also, many proofs in Statistics also rely on combinatorial arguments. Don't get me wrong: combinatorics is hard and tastes like cardboard, but it's incredibly important.

I had what I think is a pretty standard sequence in undergrad stats+probability, with a lot of counting methods up front and continuous distributions and other topics later. Somehow the counting-focused part of the course felt arbitrary and disconnected from what came later, and it was only after I had a pretty good grasp of the topic overall that I could come back to the counting-centric pieces and fit them in as a kind of degenerate case within this broader schema. I think you're onto something when you say: "I think combinatorics tends to be more relevant to toy problems than to real-world modeling." And this was part of the disconnect for me. The other thing that felt "off" about counting methods as presented to me was that they were just a bit too magical. Unstated assumptions like "every element in the sample space has an equally likely outcome" were then silently abandoned when we got to more interesting distributions (even interesting discrete distributions). So counting felt like a weird thing "off to the side" that I initially just memorized facts and techniques around.

EoP? I'm guessing Essence of Probability, but I haven't heard anything about it and couldn't find any previous posts with text that mentioned it. That said, my experience tackling new subjects had shown me that an early-on introduction to fundamental stuff is super useful even if students don't directly encounter those fundamentals directly until later. It helps frame the details in the right setting in the meantime, so they don't stumble on assumptions later. (My example: I didn't really *get* Haskell until I tried learning it again with an intro to the differences between the "logic" and "type" parts of the language was given much earlier, and I think this distinction is fundamental to jumping from the usual OOP languages I was used to.)

Even when I teach combinatorics to students who have already seen it, I am surprised at how how mysterious the "formulas" (for permutations and combinations) are to many math students (even College Juniors and Seniors). It seems that many teachers gloss over it, assuming it to be elementary or just "formulas" to memorize. So I think there is real value in giving the intuitive reasoning starting from factorial, to permutations, to P choose K, N choose K and then the various identities that usually follow in a freshmen discrete mathematics class (Binomial Theorem, Pascal's Triangle, Sum of NcK = 2^N, etc)

I agree with Ankit; as a third-year applied math undergrad i still haven't taken a proper combinatorics course, and people with less experience will have even less of a handle on such a topic. A primer video would be helpful, in my opinion.

Hmm, from my experience I think that learning combinatorics as an aspiring math student is one of the most confusing parts. I personally think that giving some weight into what the seemingly complex n choose k formula truly means would be a good way to start​ off the series. Maybe even put in a challenge problem to appeal to a more advanced audience as well. Regardless, I think a primer video would be the best way to go.

Ankit Agarwal