New probability video, chapter 1

Also I think you ought to add something about binomial or Pascal Triangle other than short mention in the beginning. Just for more curious people.

Timur Sultanov

6:30 you have a problem you say "Almost surely answer question correctly that you did not" but then it shows a counter example where question answered correctly is actually the same...

Timur Sultanov

Criticizing your content is like criticizing a Tool album. You might not get it at first, and some unlucky people might not even like it, but everyone knows that it's masterfully created with lots of thought and time, even in the case of light-hearted or casual circumstances. There's always something to come back to and learn more about after the 1st or 2nd or 3rd time watching a 3b1b video or listening to a Tool album, but the asymptotically decreasing amount of content learned per viewing/listening does not prevent me from watching a 3b1b video way more than I should or listening to a Tool album way more than I should .On top of this, both of the artistic works being mentioned are best experienced with great focus and attention being attributed to them, savoring the moment. At least you don't wait 12 years before putting out more content.

I think that with the gender thing, he was referring to the genetic probability of whether or not a y chromosome shows up, since that's where all the science of probability in that field comes in. The probability of whether or not someone identifies as male is difficult to assign a number to, where a the genetic probabilities are of course pretty cut & dry as 50/50 chances. However, I may be wrong, or this point of genetic probability might just not be what's generally perceived.

As always, well done. There's only one thing I noticed that could be improved: around 10:16 when you focus on one of the rectangles ("this rectangle over here"), I found myself momentarily confused as to which rectangle you meant, before I noticed the highlighted one. Maybe, if you faded in the highlight or "pulsed" it once or twice, it would draw attention more easily.

Petr Čertík

Hey I really love to see that you are doing the Probability Series now! I can agree with some of the critique but mostly share Jan Hönig's view. I think the over all structure of the video was nice and educating and change my perspective. I also share the experience of everyone trying to teach me Probabilities starting with everything at once. I appreciate your slower and more structured approach :) Keep it up, your work is super valuable!! Thanks for doing this

I found this link on boinboing.net the other day and it has a list of some great counterintuitive probability questions. It would be awesome to see you address some of these: <a href="https://math.stackexchange.com/questions/2140493/counterintuitive-examples-in-probability" rel="nofollow noopener" target="_blank">https://math.stackexchange.com/questions/2140493/counterintuitive-examples-in-probability</a>

Really liked the video. Probability theory is one topic where what little intuition people have often turns out to be wrong. In that sense, I found visualizing the manipulation of probabilities as ratios or partitions of the unit to be useful. That said, I would echo some of the other comments about a weaker central idea or insight compared to your other videos. While it is obvious you wanted to stay clear of the unproductive frequentist/Bayesian dichotomy, I do think that Jaynes’ view of probability as the natural extension of (Aristotelean) logic i.e. where propositions need not be only true or false, could be that insight. Reasoning about uncertainty with all its real world relevance is imho a far more motivating view compared to stylized games of chance. It also sidesteps the controversy since the uncertainty could be subjective, inherent or otherwise. I personally find it very elegant how something as pervasive as reasoning about uncertainty can be mapped, quantified and manipulated with such simple objects as the unit and different ways of arranging/visualizing it’s partitions. Not sure if it in scope or not but while on the topic of probability theory and expectations, some examples/visuals from information theory e.g. interpretation of -log(p) as information and its expectation as the average length of a message, might also provide valuable insights. Looking forward to your next video.

How about starting with something brave, like the Monty Hall problem? I would love to see it as a central guiding principle while thinking about probabilities. It confuses even smart people, and yet, it is totally comprehensible for a beginner if presented correctly. It also forces you to think about what do you expect from a concept of probability.

I really enjoyed the video. Probability is one of the topics that would greatly benefit from an intuitive understanding, which I found is often lacking from most studies of the subject matter. I would echo some of the other comments though on the central point, as the video seems to concern itself with the mechanics of manipulating probabilities. Although you clearly wanted to stay clear of the controversy over Bayesian vs Frequentist views, might I suggest Jaynes interpretation as the extension of Aristotelian logic to propositions that are not just true or false as a potential insight, with its obvious utility on reasoning in the real world i.e. under uncertainty. Then the correspondence between the manipulation of ratios and the algebra of manipulating such uncertain propositions would supply intuition as well as pave the way to the relevance of measure theory in order to make this type of reasoning rigorous. Maybe less abstract and more relevant to an engineering mindset but something to consider.

I liked the basic idea of using areas and proportions without much philosophy about what probability is. I don't know if this is more intuitive for absolute beginners, but it's certainly how I think of probability (as a measure of sets/special case of a measure space). I think a lot of misconceptions in probability come from overthinking the philosophy of it.

Hey 3b1b. I have read all the criticizing comments as well. That the video is not mysterious enough, or does not surprise or has no central point. In my opinion it's not true. You explain the basics of how to imagine an probabilistic experiment. I was taught probability at various levels throughout my education (high-scool -&gt; university) and EVERYONE tried to explain just too much at the beginning. For me stochastic and statistic was always about learning and not understanding, like the rest of mathematics. So starting it slow and laying down the ground rules, laws, and tools for imagination is indeed very useful.

Jan

Good to know, thanks.

3blue1brown

I'm not sure I understand your last point. What do you mean by "central problem"? The intent here is to use the same idea of area-representations to give intuitions for the formulas associated with conditional probability, Bayes' rule, etc.

3blue1brown

I've always appreciated your work and this video is no exception, however, Daniel and Rebecca's remark about there being no mystery or stunning revelation rings true with me. To me, one of the most fascinating aspects of probability theory is that you can use randomness to model ostensibly deterministic events like coin flips and brownian motion. We take this for granted even in common parlance about chance, but to me this feels like to sort of mystery that would give me a reason to justify a review of probability. I have a background in philosophy so I may be a bit biased here, but I also think the differences between the frequentist and Bayesian interpretations of probability are interesting, although I agree that the caveat could be left for another video. Some other things: representing probability as area is a good way to start thinking about probability in a visual way, although it wasn't clear how the 'greying of the colours' was supposed to relate to that intuition. Also, my intuition is that the 'central problem', whatever it ends up being, will need to involve more conceptual pieces (conditional probability, bayes' rule, etc.), so I would caution against confining the search space to only those problems that could be analysed with only these two concepts.

I love your work and what you do for the community of science students and teacher inspired by visual learning! I want to echo Daniel and Rebekah Slonim's points about the video lacking an inspiring through-line. I am sorry to report that this was the very first video of yours where I found myself getting distracted in the middle by a wandering mind. I don't think I can add any more thoughtful critiques than those already shared above, but just wanted to make sure that the point was re-emphasized. Thanks for all that you and your team do!

I liked this video a lot, and especially liked the point of how probability distributions are just deciding which aspect and what level of detail you view the set of outcomes with. And while I would say that there doesn't seem to be your usual style of "a revelation being built up to throughout the video", I would still say that the various geometric interpretations of the ideas behind probability are very helpful for building up intuition for a good understanding of the topic. I do have a small point to make, however: at ~1:30, you use masculine and feminine symbols as an example of binary outcomes, even though gender is not, in fact, an example of this, and the idea that it is negatively affects many people's lives. Considering that there are many alternative examples, such as up/down, on/off, or +/-, would it be possible for you to change your examples slightly? Overall, I'd say it's an interesting and usefully educational video, with your usual knack for building an intuitive understanding of the topic, and while it lacks the usual build-up to a complex idea, it does a good job of providing helpful foundations, and I liked it a lot.

Hey. Overall very nice video, but I have to agree a bit with user "Daniel and Rebekah Slonim" here. I came over to comment that while the video overall is nice, and a lot of the different parts of it are nice, I feel it's missing a "central point". Like, you often start off videos by saying something like "here's where we're going to end up: *interesting result*, and here's how we get there". In this video, it's not only missing, I'm not sure what you would even say, since it doesn't feel like it's telling a coherent story. Throughout the video, I kept thinking to myself "what are we building up towards?" and not getting an answer, which is a concern. (Still, overall great video and I'd keep most of it, including the flipping from area to bar charts which is beautiful, the "zooming in" on the tiny speck of area at the end, etc. I'm highlithinn the negative, but needless to say I'm a big fan!).

Edan Maor

Is it just me or does the video glitch/stutter at 1:00? I like most of the video, the paint-mixing analogy is good and I like the example of test scores. I think some of the animation transitions are a little too fast. I also have a suggestion for the discussion at 9:50 about "millions of sequences", I think the chosen visual could be improved by literally counting the shown sequences, i.e. showing rapidly changing digits.

Coupla thoughts: - Maybe in probability specialties, the nature of probability is a philosophical question that lots of people would raise or tweet at you about or whatever, but it doesn't distract me from your explanation in this context whatsever. I think you should dispense with the caveats about that and just do your thing. - I agree with Henry that the red-blue color blending is ugly and grey... but that's the point! the more grey things get the more the probability is approaching 50/50. maybe you could add a line saying exactly that? so that for the rest of the video when you see that color the point gets made about regression to the mean, etc. - at 5:48 I didn't get the difference between 'proxy for counting things' and 'proportions in their own right' - by 8:00 I felt like there was re-iteration and over-explanation - I liked the bit about different test scores but I thought the point that was made just before 800 was almost identical to the point that was made just before 612 - I really liked the zoom-ins to the extreme outcomes at the edge of the distribution. that makes the thesis point about the nature of probability really well. is there another visualization that could be added that offers an even clearer look at this or sharpens it? - good video!

Grant, great video, I've never seen the binomial in bar form like that before, makes me think of a hyperbolic space the way the low probability cases get crushed into the edge. My comment: your blended red-blue colors are ugly and gray! Are you blending correctly? See here: <a href="https://youtu.be/LKnqECcg6Gw" rel="nofollow noopener" target="_blank">https://youtu.be/LKnqECcg6Gw</a>

Henry Gotjen

Hey Daniel, thanks so much for taking the time to write such a thoughtful response. I think your phrase "There's no mystery, no stunning revelation, nothing to make me want to keep watching" captures a bit of trepidation I have here. You have a lot of good thoughts here, so I'll take some time to mull them over.

3blue1brown

Was looking forward to this - I seem to be in a great cycle where you release things right before I need to teach them. At 1:39, there is a faint outline of the male/female symbols still lingering below H/T - not sure if that was meant to be there still.

Quick note at 6:30 - you're making the point that the sequences can be different, but you're showing two identical sequences. Shouldn't you mix them up?

Max Goldstein

Hey, Grant, Grad student in math studying probability here. I love your channel so much that I'm currently wearing a 3b1b shirt, and I hate to be a dissenting voice here. Maybe you biased me by saying you weren't sure of the content and pacing, or maybe it's because probability is my baby, but this is the first one of your videos that I haven't loved. It feels like the wonder, beauty, and elegance in all of your other videos is lacking here. There's no mystery, no stunning revelation, nothing to make me want to keep watching. The goal of introducing probability from the perspective of a measure space and still making it feel intuitive is a creative and noble one, but I'm not sure it's the right idea here. You say that probability is hard to define, and it is, but the whole "just think of it as a proportion thing" doesn't actually help. I can just see a frustrated novice asking the question "A proportion of what?" The novice loses any intuition about chance that he had without gaining any additional clarity. Since you can't get away from the mystery, maybe you should embrace it. Talk about how a helpful way to think about probability is as the relative proportion of the time that an event happens if you do it often enough, and then discuss the limitations of this view, like the fact that you can't apply this definition to events that only occur once (like flipping a specific coin once and then destroying it so that it will never be flipped again), or the fact that the Strong Law of Large Numbers is something you have to prove, not something you can take as a definition, etc. Then, once you've discussed a few ways of defining probability and pointed out some of their problems, you can say something like, "Well, we can disagree about what probability is, but we can all agree about a few things. Whatever probability is, the probability that *something* happens should be 1, the probability of any event should be between 0 and 1, and if there are two events that cannot occur at the same time (like rain and sub-zero temperatures, or like rolling a 3 and rolling a 5), the probability that one of those two events will occur is the sum of the probabilities that each of the events occurs." Something very roughly along those lines, maybe. And maybe at that point, you can start playing around with visuals like a rectangle representing all possible weather events or all possible dice rolls, and show that "proportion" is a good way of thinking about probability because it satisfies all these axioms. But maybe all that should be the second video. My suggestion for the first video would be to present an overview of some of the many different things probability touches, the kinds of questions you can ask, and a few stunning results like the central limit theorem. Focus on arousing people's sense of beauty and wonder. That's what makes us keep coming back to your videos. And of course, a video like that would be a great time to introduce some examples (like repeated coin tosses, random walks, waiting times, whatever) that will help motivate stuff in later videos. Another possibility: not sure if this would work well at all or not, but you could consider motivating the probability space stuff by talking about the multi-verse theory. Not so much about whether it's true or not (it's not), but about how a coin should come up heads in "half" of all universes, etc. One more quick thing: you say, "The art of doing probability is choosing what level of detail you care about, how finely to chop up the area, and then understanding how one level of detail can inform you about the others." I'm not sure I agree with this statement or think it's helpful. I get what you're saying, but it just doesn't capture my feeling about what the art of probability is, and I don't think it really helps novices understand it much better. But no matter what you do, I'll always be a loyal fan of your channel!

Daniel and Rebekah Slonim

I am super happy you took up probability. i understood the video , when i watched it for the 3rd time, as i was not able to figure why you are saying these things. i was wondering if you could drive the concept after introducing a problem first. may be that would help the viewer to relate the concepts to.

Great video! One minor point: I wonder if your comment early on that the area is the 'space of all possible things that can happen' would be confusing to some. It might be clearer to say that it is the 'space of all possible things that can happen for a given experiment' as you then describe the different experiments. Different rows of the brick wall would then correspond to different experiments.

Sanjeevan Ahilan

That'll be chapter 2: <a href="https://youtu.be/AbXJW2u_rdw" rel="nofollow noopener" target="_blank">https://youtu.be/AbXJW2u_rdw</a>

3blue1brown

I really think you hit the right start to the series here. Your explanation of sample spaces (without using the word sample space, good choice) was clear and elegant. The brief switch to test scores was effective as a second perspective. The animations were simple and to-the-point, but pretty. The “paint mixing” analogy was a pedagogical headshot at a difficult idea. Really proud as ever to be supporting this.

Jacob Mirra

Really nice way to "begin" the series Grant! I think it could be cool to bring in Pascal's triangle (and all the sweet patterns that come with it). I realize that would be a bit much to fit into one video and I'm sure you've already thought of it. Just something I'd love to see in the coming chapters. Overall great vid, thanks!

Hi Grant, the whole philosofical question aside thing you did seemed a bit off to me. Also as a starter for the series translating from "random" to "proportion" kind of falls from the sky in the first minute of this video. I was wondering, will you do an intro for the series?

I think it‘s not as good as the other „essence of“ videos. I had the feeling that there was a bit too much different things going on, it was too fast for me. Maybe it is because I haven‘t had my probability course in university yet, but that is exactly why I was looking forward to this series. I am thrilled for what will come in the future

Supreme

Ha, good catch! I guess I was unlucky enough to hit the 1/8 probability of a collision for randomly generated animation.

3blue1brown

Awesome feedback, thanks so much! As to your talk, I think you should certainly put it together. Something tells me you'd be a better candidate to write it than I would :)

3blue1brown

Looks great. Not a big deal, but at 6:32 when you are comparing the two students, you imply that they didn't get the same questions correct, but they did...

Gabe

They're rescaled so that the mode is the same height to better see the shape I think. The corresponding distributions have different vertical scales.

Hi, Grant! First of all, thank you for the lecture... Probability distribution is the math subject which I have difficulty.... About the philosophical question about that probability represents a proportion.... I have a opinion as aspirant in physics... Like in Quantum Mechanics which the results are representing by a probability distribution either.... I imagine that is the Proportion of the Fourth dimension in General Relativity, it means... The "slice", or this "hologram", of infinity results, is the Time Dimension which is movement. The proportion of situations in the Fourth dimension. What do you think about this? Gratitude 🙏

Is there not a problem with the areas shown at 10:50? Should they not all have area equal to one?

I love it! In case that's not vociferous enough, I thought it didn't benefit much from the quiz example 6:13-7:09, and that the connection to the probability distribution was clearer with the coin example later at 7:25. Just my two cents, so to speak ;). Excellent work, as always.

This is in pretty good shape. Here are some impressions I jotted down as I was watching. Things I liked: * The brick wall analogy is nicer that Pascal's triangle. The visual intuition of comparing areas is much easier to access. * The visual of zooming in on the 20:0 corner is really striking. Keep that for sure. Some possible improvements: * You use an em-dash to separate the tallies as in 2 -- 0. When that first appeared I read it as 2 minus 0, and had to pause for half a second to figure out what you really meant. It's a small thing, but using a colon or some other separator might be better. * The section between 5:00 and 5:35 could be more clear. You take away the counts and leave just the colored blocks on the screen right when you're talking about the quantitative meaning of the diagram. I think you should leave the numbers up there (maybe not ratios, but counts?), and perhaps also have text on the screen emphasizing the 3:1 ratio of areas, since it's such a key point. In my style, I'd do this with little labels and arrows pointing to the blocks. That feels like not your style, but I'm sure you can find some way to hit the point three ways: visually, audibly, and with text. Some possible asides: * Maybe make an optional followup video that shows all the places that binomial coefficients show up? * I've always wanted to write a talk that walks through Bernoulli's original proof of the LLN (as recounted in Stigler's History of Statistics). The point of that talk would be to show people how hard these ideas are, and how slowly they developed historically. Maybe I can persuade you to write that talk as a video, then I don't have to write it. :)

John Rauser

I really like the idea of using surface areas to represent proportions! I wonder if it would be easier to start with thinking of a continuous surface, where you can imagine dropping something from above. Then define each point as equally likely to be hit. Since a surface area has infinite points, it doesn't tell us anything. But if you then say "imagine we split this area in two parts of equal size, and then think of how likely you are going to one of the two surfaces. Now, you can intuitively see that both are equally likely". That way you can bridge smooth probabilities like infinite coin tosses (without even introducing them yet!) and the discrete distribution of (say) 10 coin flips. Or maybe that's actually more complicated for now - but I assume discussing the distribution of doing infinite coin flips will eventually be part of the series, and then talking about smooth surfaces would be useful, no? Also, maybe it is good to briefly mention that "one head, one tail" is not the same as "first head, then tails" when justifying that you can merge the area of the latter two possibilities. Each unique sequence of coin flips is always equally likely to happen, after all, and it might make that part more understandable.

Job van der Zwan

Looking very good! The pacing feels natural to me. And visually intuitive as always. Were 'rectangles of area 1' the first representation to come to mind, or had you gone through other visual representations before arriving here? This is more of a general question I've had in mind - I've wondered it about explanations in past videos as well.

Dillon Strichman

Yay!!! Probability rocks, great work!