Why “probability of 0” does not mean “impossible” | Probabilities of probabilities, part 2

An introduction to probability density functions
Help fund future projects: / 3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: 3b1b.co/thanks
Curious about measure theory? This does require some background in real analysis, but if you want to dig in, here is a textbook by the always-great Terence Tao.
terrytao.files.wordpress.com/...
Also, for the real analysis buffs among you, there was one statement I made in this video that is a rather nice puzzle. Namely, if the probabilities for each value in a given range (of the real number line) are all non-zero, no matter how small, their sum will be infinite. This isn't immediately obvious, given that you can have convergent sums of countable infinitely many values, but if you're up for it see if you can prove that the sum of any uncountable infinite collection of positive values must blow up to infinity.
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
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These animations are largely made using manim, a scrappy open source python library: github.com/3b1b/manim
If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with KZbin, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
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[@3blue1brown]

If you're curious, I never ended up making the third part of this. Or rather, I made part of it and thought it wasn't very good. The plan is to put together something like a probability series this year, where the beta distribution will surely be one of the topics. Thank you for your patience!

[@sw3aterCS_]

And thank you so much for your hard work!

[@curiouslyglobal3538]

Thanks for the update. I will be waiting for it!

[@bora6981]

3 hours ago wow thx man

[@bacicinvatteneaca]

3h ago? Is this being recommended to everyone all of a sudden?

[@flipflipshift855]

@@bacicinvatteneaca probably because someone recently donated a bunch of money in hopes of a part 3, so he felt some responsibility to clarify.

[@Daisyboobs]

"The probability of the dart hitting the board is 1". You obviously haven't seen me play darts.

[@reinatr4848]

Or the wall

[@____-ck1vp]

Reinatr48 or the face

[@jonidepp8797]

nice one!

[@xephyre6955]

Or the person.

[@Leekodot15]

Or yourself.

[@barney2159]

Crush: You have 0% chance of being with me! Me: So you're telling me there's a chance?

[@_kopcsi_]

no. there MIGHT be a chance, and NOT there IS. it’s a logical implication between the two statements (“0 probability” and “impossibility”), and there is a different relation between another two statements (“0 probability” and “possibility”). when you have impossibility, it must be an event with 0 probability. when you have an event with 0 probability, you might have possibility and impossibility as well. all of this is due to our mathematical toolset’s limitations (infinitesimals are treated as zero). that’s why there are extensions in mathematics which can handle these situations better (e.g. hyper-real numbers where infinitesimals are not zero anymore). but when we have this ambiguity due to this kind of limitation, a “meta-probability” level emerges, since you can have possibility or impossibility when you deal with an event with 0 probability. this is a meta-possibility. ps.: well, by the end of my comment I realised that you were technically correct, since there is no difference between “existence of chance” and “possibility of chance”.

[@thatoneguy9582]

k0p1k4 alright slow down Socrates

[@ultimaxkom8728]

@@_kopcsi_ _"ps.: well, by the end of my comment I realised that you were technically correct, since there is no difference between “existence of chance” and “possibility of chance”."_ This is so sad... Can we hit 1 million likes?

[@AbhishekSharmahehe]

Well,there might be a chance of chance .

[@luna010]

@Angel Yotov No; being with me or not being with me is binary and not continuous so if being with me is 0% the only other possibility which is not being with me would be 100%. “There’s a 0% chance that there’s a 100% chance of you being with me” would be a better example.

[@fukinyouup]

4:00 "Wait, it's all calculus?" "Always has been"

[@caesarinchina]

underrated comment

[@realbignoob1886]

lol

[@choochoobob123]

Exactly what I thought about

[@parthibanpalani6490]

Made me laugh :p

[@RussTeeTrombone]

🎯

[@3blue1brown]

I have to imagine it's frustrating to follow this channel. I believe this is the third video in a row (excluding those on epidemics) that I ended by saying something like "we'll look at Bayesian updating in a continuous context in the next part". But whenever I think hard about the setup/prerequisite section of that video there's always something interesting enough to pull out to stand as its own video; there are just so many interesting topics here! Thanks for your patience, and hopefully, everyone gets that the goal here is to just hit as many fundamental ideas in probability as is reasonable. Also, in parallel with making these probability videos, I'll be trying a very different sort of experiment on the channel soon...stay tuned.

[@michaelliu8887]

It's fine because we still love your content

[@itwasinthispositionerinoag7414]

if by frustrating you mean awesome then yes yes it is

[@MayaPasricha]

Even if we get left hanging sometimes, we still love to see all your videos and experiments :D

[@ojotabe3]

Bro, frustrating is not a word I'd use to describe anything about this channel except my inability to fully grasp everything

[@enthdegree]

pls do a video on the difference between the questions that map and mle are answers to

[@adrift8871]

I like how whenever he says something, the little student pi's go like: *hmmmmm*

[@lukarikid9001]

xPureOblivion "It's big brain time"

[@bowel_movement]

@@fatihaksu1830 ?

[@anthonyontv1061]

xPureOblivion that is also what I took from this

[@Safwan.Hossain]

@@fatihaksu1830 u

[@genhen]

we are the little πs

[@minerharry]

1:37 Those are some nice decimal places you have there. I recognized pi, obviously, followed by e and then later phi; but that third one was strange. 4.6692? What kind of a number was that? That’d have to be the square root of like 19, which is a weird number. Curious, I looked it up, and - with no context - the Wikipedia page for the Feigenbaum constants came up. Wikipedia pages on higher math are completely unreadable, of course, so I looked it up on KZbin and found a Numberphile video on it, because Numberphile has a video on every single number, and - because of a tiny little Easter egg in a video that I was rewatching for the second time - accidentally learned about a completely unrelated branch of mathematics and an incredibly strange phenomenon that arose therein. I love the internet, and I love your videos

[@davids.4431]

I love MItchell Feigenbaum. I think you'll like the book "Chaos" by James Gleick that covers the story of Chaos Theory (and related stuff, like non-linearity) very well. I have to say though, it's meant for the layman, so it does not go in-depth on any of the topics. It's more of a 'review' of the scientific community at the time, and the challenges of the emerging change brought by the idea of Chaos Theory (and those who thought about those ideas).

[@luigiboy72]

@@davids.4431 wait there's literally a theory called Chaos Theory??

[@davids.4431]

@@luigiboy72 though the name is a bit misleading nowadays, it started off as the study of disordered and seemingly random behavior (be it of the weather or of certain equations), so they deemed it appropriate to call it chaos back then. Suffice to say that the badassery of studying 'chaos theory' was very welcomed by scientists in 1970, especially after a decade of fighting for it to be 'officially' recognized. Anyway, I highly recommend the book I mentioned if it has piqued your interest!

[@jb888888888]

It's literally impossible for Numberphile to have a video on every single number.

[@Mayank-tm2km]

@@jb888888888we need a video on probably of numberfile not having a video on every number xD

[@BenedictGS]

9 month later still waiting for part 3, it is okay take your time.

[@MA-kn4zm]

ME TOOO!!!!

[@warchiefoomii]

@@MA-kn4zm +10. i'm in raptures as to how this all ends

[@a.fleischbender7681]

One year and 5 days now. Still waiting.

[@MA-kn4zm]

@@a.fleischbender7681 god damn it how do we get his attention, i wanna watch all 3 in order again

[@Rubbenzito]

In Latin it is called "coitus interruptus"

[@jacemandt]

One of my favorite math jokes, relevant here: A mathematician is a little drunk, and nudges the guy next to him at the bar and says, "Hey, think of a number. Any number at all." The guy says, "*Any* number?" "Yeah, any number." "Okay, I got one," the guy says. "Is it rational?" the mathematician asks? "Ummmm...yes..." "HOW UNLIKELY!!!"

[@jasmeetk0]

That's a good one

[@justsomeguywithhalfamustac6837]

I'm big dumbness and I no get

[@henriquerock703]

@@justsomeguywithhalfamustac6837 i feel you dude,didnt get it either

[@justsomeguywithhalfamustac6837]

@@henriquerock703 yes

[@SgtSupaman]

@@justsomeguywithhalfamustac6837 , there are infinite numbers. In the infinite set of numbers, an overwhelming majority are irrational, so much so that, if something were to pick any number at random from the infinite set of all numbers, the probability of picking a rational number is 0. A good way to illustrate that the probability of picking a rational number is 0 is to just imagine that you are creating the number you pick one digit at a time with each of the possibilities (0-9) being equally likely. So, to get a rational number, when you are picking the digits that go after the decimal point, you would basically have to get an infinite amount of zeroes in a row.

[@shayanpoordian5986]

When I was trying to learn linear algebra, you put out a series solving all my confusion. then when I got interested in neural networks you put out a series which made me dive deeper and end up trying to learn stats. then you put a series on stats.

[@shakiwizao]

Shayan Poordian Oh man, it went the exact same way for me. Started the engineering grad and the Calc/Algebra ones were fundamental to my success. Then the Diferential Equations series and now that I'm into data analysis, stats. God knows how much I love this channel :D

[@arpitdas4263]

You're a lucky one. I still dream of that elliptic curves video

[@gattungswesen1630]

Grant Sanderson really is the gift that keeps on giving.

[@Aldrnari]

I kind of went from the other direction: From studying statistical science, I started branching off into more traditional mathematics, like linear algebra. His series on linear algebra is also what brought me to the channel. And I'm thrilled that he's branching into stats/probability/data science, because that's my wheelhouse.

[@Lucky10279]

@@Aldrnari His linear algebra series is quite helpful. His video on change of basis in particular is what made that concept _finally_ make sense to me. I really liked his analogy to two people choosing two different vectors as their basis and then making it into a puzzle of how to translate between the two different reference frames.

[@saurabhmehta7681]

This series really makes probability and its probabilities click for me. Hopefully the long awaited part 3 will be uploaded soon :)

[@Daniel-cc6gs]

@@mEh9003 brother it's 2$

[@stuiedaman]

​@@Daniel-cc6gs rich

[@orvinal2883]

​@@Daniel-cc6gs in my country 2 dollars will feed my family for a day. That a lot of money and take several days of work to get.

[@Daniel-cc6gs]

@@orvinal2883 where are you from?

[@orvinal2883]

@@Daniel-cc6gs Milwaukee

[@inordirection_]

Mr. BlueBlueBlueBrown, Part three? Sincerely, Probability Stans Worldwide

[@-guitarhero]

never knew his name was (Blue^3)Brown

[@inordirection_]

@@-guitarhero Now you know

[@user-ht3tp3uj4v]

Mr. Blue+Blue+Blue+Brown

[@kasskoulle]

Day 448: Still waiting. What do we do to get part blue of this?

[@LiberOpine]

Mr. (Blue+Blue+Blue)(Brown)

[@dysxleia]

Whenever he has some arbitrary, long decimal number, he sneaks in π and e and golden ratio digits. 1:37 for example

[@a.o.3523]

D-Rock good catch!

[@adamzeggai5506]

and feigenbaum constant for the second to last

[@squibble311]

π, e, δ, φ yay

[@ChristianPerrotta]

in class with my students, I like to sneak 142857 somewhere every class.

[@KiLLJoYYouTube]

He’s even on 2.72 M subscribers lol

[@nanigopalsaha2408]

1:31 If the numbers after 7 seem familiar, they are: 0 1 π e The Feigenbaum Constant φ

[@giannisr.7733]

After I saw π and e knew the other 2 were not random, thank you friend

[@malignusvonbottershnike563]

Well, considering the date of upload, it's nice to find a couple of Easter eggs in this video :)

[@dexter2392]

Good to see the Feigenbaum Constant is now considered somewhat of a famous number among math fans along with pi, e and the golden ratio. Veritasium might have something to do with it...))

[@hongkongball7101]

@@dexter2392 I like to think of it as a tribute to Mitchell Feigenbaum who sadly died last year in June

[@sadhlife]

it's sad how fiegenbaum doesn't have a universally known symbol yet

[@valeriodilecce1988]

I've been following this channel enthusiastically for years, yet I just noticed today in 2020 that the students/teacher pi creatures (2:37) are 3 blue and 1 brown. Yep.

[@zionj104]

(slow facepalm)

[@zionj104]

@Eric Lee Main comment. I've noticed it since the first video. I didn't even think of it as "noticing".

[@vendybirdsvadl7472]

IT also appears that each student got different blue. Middle student is darker and The third student is The lightest

[@timjackson9067]

Also there are 3 students, 1 teacher, and 4 pi's. 3.14 maybe just a coincidence?

[@jesusgallegos99]

@@timjackson9067 Actually, if there are 4 pi's, it means that pi=4.

[@JetFalcon710]

8:31 That reminds me of one of Zeno's paradoxes, where he says that one grain of mullet falling does not make a sound, but a thousand grains falling does make a sound, seemingly showing that many nothings somehow make something

[@emmawatson9180]

Nice

[@NoriMori1992]

Sometimes I wonder how Zeno managed to get through life, given how often he thought about the seeming impossibility of iterating a bunch of small things into one big thing.

[@mieszkogulinski168]

@@NoriMori1992 Or he was just a troll

[@thetophatgentleman4634]

My friend trying to comfort there is a chance of me getting a girlfriend.

[@cifar10]

Girlfriends are discrete objects, and in a discrete setting, a probability of zero still always means it is impossible. Sorry bud

[@er.you2594]

lol

[@NovaWarrior77]

@@cifar10 why'd you have to hurt us this way???

[@NovaWarrior77]

Don't worry man, girls love top-hatted gentlemen.

[@Arya-sm5jx]

@@NovaWarrior77 as a girl I can confirm this

[@frankbucciantini388]

People: "What are the odds..." Grant: "We gotta take a look at the probability density function".

[@mesplin3]

Unfortunately, they tend to difficult to find. Such as, what is PDF of 2 friends meeting each other at a coffeeshop and haven't seen each other in years?

[@ZackXa]

Han Solo: NEVER TELL ME THE ODDS

[@LegoEddy]

Epidemiologists: what are the log-odds?

[@absolutelyproprietary6896]

Now achieving impossible has a whole new meaning

[@irvinep]

I struggled for literally 5+ years to understand the shift from PMF to PDF and you just explained it in 10 minutes. THank you so much mate

[@Mindraker1]

"Boy, this looks like Integral Calculus..." ... "Oh"

[@realbignoob1886]

Mindraker1 lol

[@grivar]

Interesting approach. So far, when I've seen PDFs in the wild, I've interpreted them with a PDF viewer. No longer!

[@OrangeC7]

I knew I was doing it wrong the entire time!

[@muchozolf]

Asking the important questions right here

[@ZackXa]

Adobe Reader works pretty good.

[@ollerich32]

You should not have seen any of them in the wild in the first place. Printing out PDFs is just so 90s ...

[@professoreggplant9985]

Silly little distractions every time I had to write PDF on my work.. Cursed ambiguities

[@SomeFreakingCactus]

Hey y’all, just thought I’d drop by for all the teens watching this video to say that you’re doing good. Keep it up. I’m learning this stuff at uni.

[@lagrangian143]

lol imagine a teen who's watching this but he's learning complex differential geometry and just watching this for fun

[@kelly4187]

I'd love to see the third video in this three part series when it is ready! You've done a great job setting the scene. If I had explanations and graphics like this on my statistics MSc it would have been much smoother!

[@danielwilson2658]

“So you’re telling me there’s a chance!”

[@robertcameronjones]

Love it. Great movie reference.

[@gepisar]

“The world in which you seek to undo the mistakes that you make, is different from the world where the mistakes were made. You’re now at the crossing. And you want to choose, but there is no choosing. There’s only accepting. The choosing was done a long time ago.” ~ The Jefé

[@cube2fox]

"The probability that I would date you out of continuously many possible guys is 0!" "... so you are telling me there is a chance? :D After all, according to 3blue1brown, probability=0 doesn't mean impossible!"

[@dannyboywhaa3146]

I coulda been a contender, Charlie?

[@xyzct]

Probably.

[@leftfootfirstpolitics]

8:38 probability the dart hits somewhere on the board is 1 With my throwing arm, that's pretty generous...

[@trickygamer555]

me: (misses something literally next to me) also me: I blame quantum physics and whatever things are unexplained.

[@lucaslucas191202]

1:55 I think this 'paradox' is a good example of what infinitesimals can be useful to describe. A value which added together infinitely many times gives something finite. It's a concept that is hard to understand but is clearly a thing for these sorts of concepts. Not that it gives a useful value, but it's a useful concept to understand this.

[@MrNightLifeLover]

Love this video: I have been introduced to PDFs like "there is a thing called PDF, learn it" and never really thought about the underlying concept. Thanks for the awesome video!

[@luxeproultimate360]

The Carlsberg slogan "the best beer in the world, probably" takes on a whole new meaning now...

[@Poodz_]

There is a probability of 0 that Carlsberg is the best beer in the world.

[@lazergurka-smerlin6561]

@@Poodz_ Except there is a discreet number of beer manufacturers so there's a small, chance

[@TheGlassgubben]

@@lazergurka-smerlin6561, but it's Carlsberg, so it's zero anyway.

[@shakofarhad876]

Just a former maths teacher talking into the internet void about probability: To me it makes sense that the dart has a probability of 0 of hitting a specific point on the dart board. If you are aiming at a specific point, it means that you are betting on the fact that your accuracy will be on the level of atoms, and even smaller (because math has no Planck length). You literally are boasting infinite accuracy, which is impossible. That is why your probability of hitting that specific point is 0. But if you say "I am going to hit Bullseye". Then things change, now you are being reasonable. The bet is no longer on hitting the infinitely small point, but rather hitting an area which contains infinitely many of these infinitely small points. In some sense you have infinitely higher probability now since you have infinitely many small points. But of course in our real world we have the Planck length which means that we are never really talking about infinity, just very big or very small numbers. That also means that the probability is never truly 0, however it is extremely tiny. ^^

[@WolfrostWasTaken]

This comment deserves more likes. It made me understand the matter at hand even more. Props!

[@jesusvera7941]

oh, so thats what he said? pretty obvious

[@flyingface]

Nice analogy But I think you're somewhat conflating physics with math here

[@nanigopalsaha2408]

Well in fact we don't even know whether Planck length is the smallest unit of length. We *think* so. There is no proof of this.

[@zakthesquirrel7621]

what if we use a plank length dart ?

[@ibraamnashaat5584]

It has been an amazing series. Part 2 was published 2 years ago. Are you going to prepare part 3 anytime soon?

[@TarunJangra16]

I learnt this in my engineering and I can assure this is extremely fascinating.... The different distributions, the applications, the hypothesis, LOVELY

[@josephv4174]

3b1b, How about taking consideration on making *"Essence of Number Theory"* ? Much respect!

[@Nylspider]

I would love that tbh

[@varunraju1569]

That would be amazing, but is there much scope for visualization?

[@sohampatil6539]

Yes great idea

[@infinitymatrix2890]

That would be really cool

[@winoo1967]

This is a great idea!! I intend on presenting myself to the math Spanish olympiad next year, and it would be really useful

[@magnuseifr]

1:31 that's some sneaky famous constants right there

[@jpe1]

Magnus Eide-Fredriksen Easter Eggs in a video released on Easter... who woulda thunk 😀

[@GeeTransit]

@@grammarnazi9456 *thunk*

[@thorr18BEM]

Denislav Ivanov , if you don't like colloquialism, then why not also correct the use of *woulda* which is obviously actually *would'a*, a corruption of *would've*, a contraction of *would have*? Why would it be OK to corrupt one word but not the other?

[@squibble311]

π e δ φ

[@nostalgiafactor733]

I don't get it? please explain. I don't see Pi, e, etc.

[@juan.jose.111]

its so brilliant and structured the way you explain really hard concepts. i've read hours and hours and never had this concepts clear.... now with just this video. i'm sure it will stick in my mind. tks so much

[@DockedSlinky]

Your job seems awesome. You get to make videos for others to learn while going really in depth and learning a lot. That’s the dream

[@itwasinthispositionerinoag7414]

there is a probability of 1 of me watching this

[@myreneario7216]

Did you watch it?

[@hongkongball7101]

Lmao agadmator is here too? Check check check check

[@itwasinthispositionerinoag7414]

@@myreneario7216 I did, but now I realise that I shoulda drawn a curve taking unknown effects into account like internet going down or me randomly dying half way through...in conclusion I lied, sorry!

[@shayanpoordian5986]

@@hongkongball7101 captures, captures and then captures

[@yaroslavpanych2067]

@@hongkongball7101 FU, this is not agadmator!

[@user-he1rn5uu5w]

Hears "probability of a probability", recalls "slope of a slope" Me: hmmm....I don't think I like where this is going.....

[@technoultimategaming2999]

Differential Calculus?

[@romilgoel4191]

ヘ（。□°）ヘ

[@lifestyleastherapyafterstr9423]

Google can be so annoying sometimes, if you google "slope of a slope" you get no relevant results

[@alvaroalejandrollanos9139]

Excellent way to explain something that most of us studied in Statistics class but took it for granted instead of questioning the intuition behind the integration in density functions.

[@Nvenom8.]

I really do need that part 3. I'm encountering some problems like this in my dissertation work, and I'm very sure that your next section would cover what I need to understand, but I've found no sources that put this topic in language I can borderline understand other than your videos.

[@CaptainSpock1701]

5:26 - And here I was looking for Acrobat Reader every time I saw a PDF?

[@shivanshsharma6921]

Hahahah

[@shivanshsharma6921]

Me too

[@shivanshsharma6921]

Hahaha Me too

[@dramforever]

The way I think about this 'paradox' is: Nothingness does not have a *size* . A point does not have a *size* either. But the latter can still exist. The introduction to probability density function is fantastic, but density is everywhere! For example, if I have a piece of sugar crystal (that I assume to be a continuum of matter with uniform density), then as I split it into smaller pieces without deforming each piece, the mass of the pieces gets smaller and smaller, but the (mass) density is the same. If I pick an arbitrarily small piece, trace back to where it was, I can say that the crystal had such and such density there. In fact, this is what I meant by uniform density -- same density at every point inside. We have successfully associated a point of matter with a mass density, even though it has no mass.

[@Fera-gr5mm]

We just mentioned in the class we are not gonna talk about infinite probabillity spaces, and went on.

[@thoe4503]

I'm of the "school of thought" that says while a point is supersmall, 1/infinity perhaps.. that's not quite 0. An infinitesimal number isn't 0. It's also a matter of convention and notation. E.g. it's widely known that 1/3=0.(3). Many people use this to say 0.(9)=1 which would be in favor of 1 point having no dimension. But the truth is 1/3 is not actually 0.(3) it's widely accepted that it is, but it's actually not. 0.(3) is a recurrent definition that keeps adding 3 at the end until at infinity the number *tends* towards 1/3. It would be more accurate to say lim(0.(3)) = 1/3. In the same way it's a matter of convention to accept that points have no size, no mass, etc, when in fact this is an approximation so that we avoid perhaps exotic infinitesimal maths.

[@orlandomoreno6168]

A point does have a size: 0

[@thoe4503]

@@orlandomoreno6168 or maybe a point has size 1-0.(9) which some people convene to be 0? :)

[@trickygamer555]

@@thoe4503 A question I found: is 3/3 the same as 1, or is it 0.(9)? well, if you take 6/3 it becomes exactly 1.(9) thus, 0.(9) = 1 Don't tell me wrong, I dislike being told wrong if the cause is an infinitely small mistake.

[@abin_._antony]

I had been puzzled the by how I should link discrete content vs continuous content. You cleared it very well in the video. Thank you

[@DrWonderVoll]

It is fascinating how you can explain such complex topics visually!!! Kudos!!!

[@johnchessant3012]

I just re-watched the "divergence and curl" video and now I can't stop thinking about how the factory at 2:25 has positive div.

[@JoeyFaller]

ah yes,the car field xD

[@rcksnxc361]

Bruh

[@dexter2392]

Vector car field

[@joshuastucky]

Great video. As a math PhD student myself, I always enjoy your content. You give great explanations to the non-expert, but include the proper references and side-comments that appease the folks that do this for a living. Kudos for learning about measure theory and Lebesgue integration; that stuff can get quite technical.

[@namehkoudsie6075]

Dude you got me out of a break up when I was 20, and again at 26 These videos are the best product of 4 centuries worth of science

[@pedrorrivero]

When will we get the next and final chapter to this awesome series? 🙃 keep coming back hoping to find it haha

[@jasonremy1627]

I know. Just had the same thought myself!

[@azophi]

My friend: dude you have a 0% probability of getting her to go on a date with you. Me:

[@maythesciencebewithyou]

infitessimally close to zero, is not really the exact same as zero. It's still above. Something that is truly impossible will have a probability of exactly zero.

[@azophi]

@@maythesciencebewithyou haha wouldnt it be funny if someone made a joke about that? (also calculus exists lol)

[@randomsnow6510]

Gets differant girl intead

[@moyrml]

If I look back to the first time I studied this - first year probability course - I feel like I missed on a REALLY important course. It was difficult for me, and a "small" course in terms of points, so I was ok with not fully understanding things, getting a low grade (but passing) and moving on. If I could go back to younger me today, half-way though my masters, I'd probably say to myself "This is important, try harder you donkey". As usual, great video!

[@EvanZamir]

moyrml It’s super important. A great book on probability that I’m going through now is Blitzstein and Huang’s Introduction to Probability. It’s a fantastic textbook and if you go through the problems (many of which have solutions online), you’ll learn a ton.

[@nonamehere1626]

I've been dealing with probability density functions in physics a lot for 3 years, I've started this video like I start most 3b1b videos thinking "I might learn a useful interpretation or two", ended up thinking "Damn, I actually don't understand shit". Will know seek some quick measure theory notes online to see what's what. Once you get gritted with the repetitive math you forget to ask about the greater picture.

[@HorukAI]

I'm always late five years with understanding in my life, but then I understood that if I lose that feeling it means I didn't progress. eg. I was really lousy math student in retrospect, I got the degree but should've appreciated the process of studying more. Then life happens (12 years passed) and the time window for the serious study is gone even though my girlfriend and I are going through textbooks of most interesting fields together over weekends (for us - ZF set theory & functional analysis )

[@LukePluto]

same

[@cookiequeen5430]

Same for me! Our probability course felt so stupid

[@666MrGamer]

Hey Grant! This video is amazing. I was wondering if this series would ever come to and end with the third part with beta distributions? I am really curious. Thank you for your great work!

[@DJStompZone]

(Partially in response to the pinned comment by OP) Can we all stop and thank and/or applaud 3Blue1Brown for the lack of sponsored content? I really appreciate the integrity and the obvious desire to drive the channel in an "educational over commercial" direction. Regardless of anything else, it's a very entertaining channel and I, for one, greatly appreciate the purism and consistancy. *Steps down off of my soapbox*

[@Aqua-gf9vg]

5:25 "Anytime you see a PDF in the wild" The writing in this channel is underrated

[@eliavrad2845]

"Seeing a [mathematical concept] in the wild" a a standard Idiom of lecturers, funny lecturers, and lecturers who only think they are funny.

[@pequalsnpsquared2852]

@@eliavrad2845 I think it's the sort of thing that used to be a joke, but now isn't - people hardly realise what they're saying might be 'funny' to some

[@ganaraminukshuk0]

My first thought was converting to PDF like it's a religion. Recall the google search meme that said "how to convert to", and the search options included various religions and then pdf.

[@huhneat1076]

"The probability of a dart landing on spot X is 0. 0 probability for all the spots means a 0 probability the dart landing on the board at all." And then the dart lands on the board Thanks for this video I was so confused

[@user-vp8zx9ys6t]

Don't worry. There are just too many many points on which it could land that the sum of their zero probabilities exceeds zero :D.

[@anthonyluo12]

@@user-vp8zx9ys6t *confusion intensifies*

[@blauesserpiroyal2887]

Its kinda like infinitessimals

[@pedronunes3063]

That's a number as close to 0 as we want. But we sum everything up the result is not 0

[@truedhonifan922]

As there are infinite points and 0*infinity is indeterminate so there is always finite chances of it landing

[@meinbherpieg4723]

Please make a part 3. thank you for everything you have done over the years

[@dudamesh9541]

Getting recommended this cuz of speedrun drama

[@uncomfortablyquiet7780]

That drama is boring. Like bro really over a fake speed run? If he cheated then take away his spot and forget about it.

[@thuliumberiliumcalcium1238]

@@uncomfortablyquiet7780 not really tho, because it isn't just about a speedrun. It's taking away the credibility and authority of the ones in authority from a whole entire community. It basically proves that someone with fame money and power can do whatever they want to do and just not bear the consequences, or have them lightened

[@isaacmillen8789]

@@uncomfortablyquiet7780 Bruh

[@SquidBeats]

Jesus Christ is God and is the only way. Hell is real whether you believe it or not. I used to watch wicked anime, mastberate/porn, vape, beer, violent video games and now I don’t do that anymore; I didn’t even think changing how I am now. There isn’t multiple ways; catholic, muslim, etc will lead you to hell and I was a catholic before! trinity is a lie there is one God. Go and read the KJV, other versions gonna deceive you.

[@KingOfMalevolence]

@@SquidBeats What made you think this way? I'm curious.

[@davidargles]

This is brilliant, thank you! I'm really an engineer rather than a mathematician, but this worked perfectly in clarifying a problem that I didn't realise I had. And it's helped me to a much deeper understanding of the issues. 👍

[@clray123]

It's just an example of how mathematicians can shoot themselves in the ass just using words. If you introduce grand concepts such as "infinity" (which in any real context just means "a looping algorithm", "an iterative process" or something like that), and when you take them to extremes, that is what happens. These silly games have very little relevance to the real, finite, discrete world, that you can measure and experiment with, however.

[@ahmedabbas2595]

This is just truly beautiful! I've always had the same question and never found such simple and elegant explanation, really from the bottom of my heart, THANK YOU!

[@chetanyamishra2916]

man you know this video is incredibly useful for understanding maxwells speed distribution in KTG if you just relate it and that's the one thing I adore about this video thanks grant🙏

[@nindou6]

Very impressive animation and interpretation of the statistical concepts!

[@abhay_hegde]

When I was just wondering what to watch during this boring period of quarantine, the notification bell rang. This made my day.

[@arbs-5164]

We can all relate

[@aonodensetsu]

well, it filled exactly 10 minutes of your life

[@matheuscastello6554]

these videos make me feel like 3x smarter and are such a better way of teaching than what most colleges or universities usually do, thanks as always for this quality content :)

[@erroneum]

I remember learning statistics (after learning calculus) and seeing all kinds of connections with statistics and boolean logic, completely separate from what was being taught. The notion of individual values having zero probabilities never bothered me, but I might be a bit more willing to accept the possibility of infinitesimals than some people would like; there's something I find deeply satisfying about how you can add any finite number of infinitesimal values and still get essentially 0, but add an infinite number of them and potentially have something finite.

[@peterrosqvist2480]

Holy crap that is crazy! Could you give a demonstration?

[@dorol6375]

The integral​@@peterrosqvist2480

[@cakeyeater7392]

I feel kind of proud that I recognized that an integral would be useful in a continuous setting. Especially since it was before you mentioned it in the video.

[@hongkongball7101]

Having a daily 3blue1brown upload is not impossible, it just has a probability of 0.

[@rocketpig1914]

No, it's impossible.

[@thelolminecrafter7830]

When you started explaining about how we should view *h* as ranges, my mind immediately tried integrating the function P(P(h)). I'm starting to get worried that I've been studying Calculus for way too long.

[@maxlovell]

Love your videos, you've saved me many a time! Would really appreciate part 3 about now, if you can!

[@user-hg1ey5ec8m]

thank you i needed this for my probability class, it helped with continuing onto Moment generating functions

[@JosefFurg1611]

Having to study this on my own under full lockdown, I thank God I came across this!

[@xXDarQXx]

This is probably gonna go unnoticed, and I'm sorry if 15 y/o me is too stupid to understand these concepts but I found that your videos recently are hard to follow: in the beginning of the video it is unclear the conclusion you want to draw at the end which makes it harder to follow on the bits that need following and which bits are essential to grasp the underlying premise. Whenever you post a new video I watch it twice only in the second time do I see why those "loose threads" you've been pulling off tie up to the conclusion that you've reached the first time I watched the video. Great content btw; unmatched quality anywhere else. And I'm not sure if this is just me or not but I just wanted to put this out there.

[@3blue1brown]

This is good feedback, thanks for letting me know. It may be because these videos all began as one script which was way too long and multifaceted to be a single video. So what you're probably pointing out is an artifact of chopping up one long lecture into several short ones, and chopping down that script likely could have been done more skillfully.

[@xXDarQXx]

@@3blue1brown that can be the case. I also thank you for replying , some teachers don't even take the time to look at feedback. You are truly a gem in a million.

[@gabrielmello3293]

I'm 19 today, currently studying engineering and have been watching 3b1b for at least 3 years. I can confirm that back then, I didn't get a lot of stuff from his videos and lots of times didn't even know I was missing something. Before I make the next claim, I want to point out that by that point, I had already been really into math for at least 2 years and had been watching hundreds of hours of numberphile, mathologer, matt parker and so on, so I think it's fair to say I wasn't "new" to math. That being said, I do believe that to get a good grasp of the concepts conveyed in these videos, you need to roll up your sleeves and solve some problems in an abstract setting, and then re watch the video. I'm not entirely sure why, but all the times I've done it, it's been the most effective way for me to get into the finer details of these lessons. The clearest example of this to me was with the series on linear algebra. I had watched it 3 times before I'd finished highschool, but some things about it just never clicked. As my first semester in college went on, I watched it one more time and tons of things started to come to light. This is just my personal experience, but I think it's definitely worth a try, especially if you think that you may study something math-related in your future.

[@hailmary7283]

There is nothing wrong with reading (or watching) something more than once to get it. Not everyone understands everything at the same speed so don't feel bad about not understanding something the first time through. You're only 15, but as you get to higher level mathematics, sometimes you will have to read the same passage multiple times to understand something. So I certainly don't think you are too stupid, I think it is just the nature of higher mathematical learning.

[@xXDarQXx]

@@hailmary7283 not too stupid? So stupid but not that much?

[@rafaellisboa8493]

excited for the next part, I love this channel so much

[@johnnyjohn8428]

Hey, loved the video! Hopefully part 3 comes out in the next few days as my exam is on the 4th of May :)

[@noah9942.]

loved your livestream just now

[@connorspies1842]

Noah Blackwell haha I’m so confused... did he get hacked??

[@nonconsensualopinion]

Perfect. I was just trying to figure this out last week.

[@Philcotigo]

As always, I wouldnt understand half of it without the visualization. Thanks for all the effort.

[@Ryokusei2]

This is an excellent way to introduce the notion of continuous random variables

[@Paggii]

This sound always put me to sleep, but I want to watch it.

[@DavidRichfield]

I usually watch the videos twice. Once at night while going to sleep and again the next morning to follow and understand.

[@benzmansl65amg]

Yes!

[@marcim5172]

"He's going to use calculus isn't he?"

[@larianton1008]

You are amazing!! Thank you so much. You have such a intuitive and well defined understanding upon this issue, that all of my body's tension just melted away after watching this video. I was so stressed about understanding this, and you, my dear sir, have given me that. You are a true treasure!❤ I would hug you, if I could :DDD

[@fafufafu5784]

I have to thank you for the content since the schools are closed this actually helps me getting over the test at the end of the year

[@Arya-sm5jx]

Well time to call my high school teacher.

[@JohnDoe-ki6yd]

If they can't be nonzero, and they can't be zero, we simply extend the problem to the surreals and make them equal to the infinitesimal times a scalar, so they all add up to one.

[@revenevan11]

Yes! I love this way to explain/express this using surreal numbers! Thank you so much, I'm going to cite this example from now on when questioned on the "usefullness" of surreal numbers.

[@tejing2001]

@Jake P Technically there are multiple different hyperreal number sets, of which the surreals are the largest (though the surreals are a proper class, not a set, but I'm getting off topic now). However, any of them should allow this. The differences between them don't really come up so long as you're using them for calculus.

[@mbrusyda9437]

@@modnarsarhp it does have a value, zero is the value

[@mbrusyda9437]

@@modnarsarhp no, I mean the value is zero, not some small nonzero number

[@JohnDoe-ki6yd]

@@tejing2001 Do you know of an example of a person calculating Bayesian probability using this approach? I'm very interested in it, but don't really know where to start.

[@mihirkolli9509]

Crazy, I just took my econometrics final on exactly this topic: random variables, sample distribution, measure theory, and next time I check my KZbin home page I see this video !

[@_wetmath_]

5:26 show this to someone without context

[@actuallyloogames]

Girl: there's no chance I will be your girlfriend 3Blue1Brown: So ur telling me there's a chance...

[@mrahua]

I'm a bit too stupid to understand this, still you deserve a like for the effort.

[@Fourside__]

I feel you. Still enjoy it when my brain tries to understand 🤷🏼‍♂️

[@ankushgarg1825]

Just a reminder to kindly upload the 3rd part. Eagerly waiting!

[@CypElf]

Math teacher : "You can see here that the probability is 0, so this case will never happen" Me : "Well yes, but actually no"

[@drjoriv]

OMG this guys is so good at everything that he does. He is even self aware of the things he teaches. This whole time I was saying "this is just an integral" and responded at the end of the video by saying "the video would have been shorter if he said integral from the start" lol.

[@ras_krystafari3333]

an*

[@drjoriv]

@@ras_krystafari3333 It was based on how I said it in my mind haha. I could have said it wrong lol

[@johnschmidt1262]

The 1st part of this video reminds me of my interpretation of Zeno's paradox. What the halving of the distance represents is the "impossibility" of passing through every point upon a path. When an object travels along a path, It must somehow pass through every point on the path, yet it spends exactly 0 time at each point. Further consider the question of which point it passes through right before it reaches distance one, it's not .999 repeating, so what point is it?

[@trigon7015]

In real life, the Planck length is the smallest distance you can “travel”. In the hypothetical space, mathematical and physical laws seems to work differently. That’s how I interpret it, at least.

[@tejing2001]

Regarding that last question, a set being totally ordered no longer means points have to have predecessors/successors in the case of infinite sets. There is no "point just before 1". As for Zeno's paradox, the most compelling response to it that I know is this: Why exactly are you assuming that space can be infinitely divided, but time can't? I could imagine both being infinitely divisible, or neither, but having them be different just seems silly.

[@johnschmidt1262]

Of course Zeno's paradox has some kind of solution, Or we couldn't move at all! I'm not sure that the Planck's length thing is correct as math itself would still have this underlying paradox. More kind of a mathematical brain teaser. If you're curious about what I believe, I agree with Saint Thomas Aquinas, basically that space is primary and the concept of points arise secondarily once space exists. Space defines the existence of points but points don't define space, In other words a line simply isn't a bunch of points strung together.

[@johnschmidt1262]

@@tejing2001 I think you misunderstand my explanation. I'm allowing both space and time to be split into infinitely small parts. The argument is more like, because traveling along a line means passing through every point in order what is the 1st point you pass through have after you leave the origin? Because every point on a line has a valid coordinate why is this an unfair question? Clearly there is some answer but what is it?

[@tejing2001]

@@johnschmidt1262 The space and time thing was a response to Zeno's paradox itself, not a response to you. As for the "next point" thing, I already said it pretty clearly. Just because the points come in order doesn't mean there's a "next" one. That logic only applies to finite sets. What's the "next" real number after 0? For any positive real number x you suggest as "next", x/2 would come before it, contradicting the assumption that it is, in fact, "next". That's a proof by contradiction that no number is "next".

[@AnujKumar-qm1vw]

Eagerly waiting for part 3 of this series

[@rediculousman]

You should do a video on relating probability and frequency analysis. Remember my lecturer teaching us about the Fourier transform/series, Laplace transform and then probability. He treated them as different topics, but in the last lecture suddenly manipulated the equations to relate one to the other. It was a while ago, but I seem to remember that a sample is equivalent to an individual frequency and weight and the signal was the population. Was super cool to see!

[@lanceareadbhar]

George McFly: I'm your density.

[@Chek94]

'I'm your Radon-Nikodym density.'

[@Neoplasie1900]

This may sound a bit weird, but it's nice to see stochastics from a mathematical sound point of view. I distinctly remember that back in high school, I loved math. I loved calculus, differntial equations, linear algebra. But stochastics never really clicked. However, my fellow comrads in the same class that usually struggeled with math had absolutely no problem grasping stochastics. It took me some time through my studies to find a solid basis for stochastics and probability that actually made sense to me. And it's cool to see well explained on KZbin again! Long story short, your making some good content! Keep it up.

[@sam3oq980]

It may not look like it, but Stochastics is actually one of the hardest math concepts and requires a lot of very technical knowledge to truly understand what's going on. Heck, at my university Measure Theoric Probability is a master-level course. It's no wonder if you struggle with it. Might even say it's easier if you're not into math as you then don't have to worry about how it really works.

[@silvory7021]

Integration actually makes a lot of sense in this context. As the number of options approaches infinity, the likelihood of any one option will approach zero because there are so many possible options that the probability of any one specific option is seemingly impossible but still feasible by definition

[@DemonDaze251]

Great video! I had the same confusion between the Discreet and Continuous context

[@sophiemoser1752]

Me: *starting this video* Brain: you can stay, but I'm leaving. Me after ten minutes: well, you missed a great video!

[@edskodevries]

I remember you making that comment about darts hitting a particular exact spot being possible yet having probability 0 in an ancient, I think it was Numberphile..?, video, and it blew my mind :)

[@phookaziz3]

I applaud what you did during the lockdown months to help math stay in students forefront. That surely did need to be higher priority than going down a bayesian rabbit hole. Eagerly awaiting the next in this series, and the eventual conclusion that brings it back to the probability of having a good online purchasing experience. Would it be more useful now that we use Amazon, and other online sellers way more than we did pre-pandemic?

[@theopoldthegamer4284]

8:25 Just seeing this little guy brightened my day! Especially when he put his hand on his eye.