Binomial distributions | Probabilities of probabilities, part 1

Part 2: • Why “probability of 0”...
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/beta1-thanks
John Cook post: www.johndcook.com/blog/2011/0...
------------------
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.
------------------
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
Various social media stuffs:
Website: www.3blue1brown.com
Twitter: / 3blue1brown
Reddit: / 3blue1brown
Instagram: / 3blue1brown_animations
Patreon: / 3blue1brown
Facebook: / 3blue1brown

[@Aquanistic]

Sooo.... who is going to write a Chrome Extension to do the math for us when shopping on Amazon?

[@ronswanson195]

I am wondering if Amazon doesn't already do something like this when ordering by pertinence...

[@ChrisBryantMusic]

I like this idea. I think I’m actually going to do it

[@SebSharma]

@@ChrisBryantMusic Could you share it if you do

[@grantstenger6182]

This should be a more highly liked comment, such a cool idea!

[@agamkohli3888]

this is brilliant. I’m gonna make an Opera (my browser of choice) extension for this.

[@tyrelllaszlo6024]

Also a good approach: reading the 1 star reviews and checking for grammar/spelling and signs of stupidity.

[@Cathowl]

My favorite tactic is to read the 3-star reviews and see what people who partially liked their experience and partially didin't had to complain about.

[@Thmyris]

I always do this. Definitely worth spending time on.

[@andrewdunbar828]

It's worth doing that for the top reviews as well as the bottom reviews. "It's great!" and "It sucks" with no details are equally useless reviews.

[@alessazoe]

I tend to read the low ratings and check against what I want from the product. If they only rant about stuff I don’t care about or don’t affect my use case, a product with less positive reviews than another might still be the best option for me. Especially great tactic when the whole range of products you can choose from is rather inexpensive plastic probably crappy quality stuff from china and no high quality product with a higher price and/or higher standard production place at all. Or there are some, but they have especially bad ratings since buyers are more pissed off from a product not being decent when they paid more for it while you would expect a bad experience when you buy cheap.

[@LordEvrey]

Also sampling the worst reviews and figure out how plausible they are.

[@JohnnyJackPompolla]

Your animations have become astoundingly good and nuanced over the years. The way all the (50 choose 48) outcomes were displayed, starting slow, going faster in the middle and then ending slowly again... THAT'S a satisfying detail.

[@eccentricOrange]

That's a great observation (and yes I do really appreciate the detail and easter eggs that Mr. Sanderson puts into his videos, in various forms and not just in animations) but it's a fairly common detail. In fact, Google advises using 'easing' for their Material Design elements ( material.io/design/motion/speed.html#easing ). Normally I'd not bother pointing out a MINUTE DETAIL to somebody (like that their observation is fairly common) but you clearly have an eye for detail so I think you may enjoy learning more about it... :)

[@charusingh2159]

This channel is a gift to math community, never seen a better explainer than Grant.

[@saminchowdhury7995]

to humanity as well

[@jamez6398]

"Which one of these are better? Here's a three part series to answer that question."

[@CreepyMagician]

At least we can be thankful that we got a practical strategy without having to wait. :-)

[@nonamehere1626]

RIP me taking advantage of sales.

[@Beny123]

CogitoErgoCogitoSum he is the result of a very traditional system of education. The difference seems to be how rigorous his training was.

[@cevs3123]

@CogitoErgoCogitoSum Burger King, McDonald's or neither?

[@cityuser]

We got the answer, the three part video is about "Why is this one better than the others?"

[@kostantinos2297]

"Let's choose a random number": 0.42 Certainly not deliberate! Also, appreciated the Tesla pun with the cars. "Nikola" lol

[@wodddj]

what's the reference of 0.42?

[@tejing2001]

@@wodddj Hitchhiker's Guide to the Galaxy. The answer to life, the universe, and everything is 42.

[@wodddj]

@@tejing2001 Thanks

[@niklasl4180]

Maybe just 420

[@ganaraminukshuk0]

The ultimate irony is that Nikola Motor Company and Tesla Incorporated are two separate entities. Seriously.

[@subasurf]

I did my masters in statistics an I still find these types of videos invaluable to refresh my own intuition.

[@CodamATW]

As a math major, I find these videos nice, but I feel like the proof behind the binomial formula gives me more insight than the «interpreting the formula» part does. How do you relate to that?

[@MathPhysicsEngineering]

For those who are interested in detailed analysis and proof of the binomial expansion formula, I would recommend looking up on my channel the video called:" The Binomial Expansion Formula Derivation and Proof"

[@benwinstanleymusic]

Your level of clarity when explaining stuff is so refreshing! Your videos are a pleasure to watch, I’m excited for parts 2 and 3

[@erick_ftw]

"Which one should you buy from?" The one with the prime checkmark lol

[@wesleymays1931]

The one with 100% and 10 ratings because it's 3Blue1Brown Publishing.

[@UVjoint]

I've actually started doing the opposite. I don't have a Prime account, but I noticed that if I buy items with the Prime checkmark and choose shipping at a normal speed, Amazon just ships the product late and delivers through its Prime delivery subcontractors with the horrible working conditions.

[@chrishughes3405]

don't forget amazon doesn't pay taxes like the rest of us

[@govamurali2309]

Lol

[@DanteKG.]

@@chrishughes3405 what

[@ardiris2715]

When I see ten perfect ratings, I assume ten friends of the seller bought the item, gave the item a perfect rating as "Verified Buyers", and then received refunds offline.

[@tim40gabby25]

I'm reminded of recently handing in a bunch of positive feedback - the receiver assumed it was a cherry picked subsample entirely because of the uniformity. Matters of assumed selection bias are tricky.

[@AlanTheBeast100]

People have to start somewhere. I used to sell used books, CDs and DVDs on Amazon and got nothing but good ratings from the start to end. Then I ran out of books. Which was the plan.

[@ardiris2715]

@@AlanTheBeast100 That is no reason to game the rating system.

[@Rekko82]

Sellers with 10,000 feedback with 99 % positive rating are quite likely to cheat because they already have lots of positive ratings. Sellers with few ratings wanna give the best service.

[@obi-wankenobi9871]

@@Rekko82 Sellers with tons of good reviews have a lot to loose though.

[@stephenmcateer]

I'm dying waiting for the sequel to this. Dying.

[@akshatsharma2299]

There already is a sequel

[@briholt100]

@@akshatsharma2299 where is video 3? I've seen #2.

[@KillianDefaoite]

I am a second year applied and computational mathematics student and I can say without a doubt probability is one of the most confusing and counterintuitive areas of mathematics. I can't wait to see how you approach Bayesian statistics in part 2. This video, like all of your others, was very enlightening and thought provoking.

[@toshb1384]

"i'm not going to make a probability series" - 3B1B

[@dlahouss]

That statement is unlikely

[@dan00b8]

"It's possible, but with probability 0" - 3B1B on Numberphile

[@petervilla5221]

@@dan00b8 That must have been a rounding error due to inaccurate measuring instruments.

[@sambishara9300]

@@petervilla5221 Now I think about people as measuring instruments for their opinions. Thanks.

[@macronencer]

It's a chance in a million, as Pratchett would have said.

[@3blue1brown]

Part 2 will be out soon. I'm going to implement changes based on supporter feedback, and in the meantime am also working on a video simulating epidemics. Thanks for your patience, and stay tuned! We'll also talk more later about how to address the ways this obviously simplified model differs from reality. First, we have to establish the basic building blocks to work with. Some people have asked about if the smaller graph around 10:40 should actually be much taller, since “the area under it should be 1”. If it was a distribution, this would absolutely be true, but those two graphs are simply functions, where s is a variable, not probability density functions for s. To see how that pdf comes about, that’s where Bayes’ rule comes in, to be covered in part 2.

[@Antediluvian137]

Edit: I'm wrong. I was missing that you could also plot a z-axis with different data outcomes (478/500, 479/500, etc). The integral of that 3D plot should still a probability of 1, but the 2D slices don't need to. Isn't it because your plotting function in the video is discrete? In other words, is it rendered over a discrete set of values? Maybe there's a very sharp spike at a specific value, so plotting it continuously should still yield a total area under the curve of 1, shouldn't it? Because the total probability of all outcomes must still be 1, or what am I missing?

[@dab-jacaylofficial762]

Will you proceed sir Your serious video how long When you stop all is my curiosity Please proof us trignomatery

[@iveharzing]

@@dab-jacaylofficial762 what

[@Katherine_Zheng]

@@Antediluvian137 while the value that we assign each s to is a probability, it is just that, it does not describe a distribution. For another example: let's say we have a n sided die with 100% probability to land on one. And n is greater or equal to one. And we do the same thing as in the video, we graf the value of the probability that die lands on one as n changes. We find that the graf is just a flat line at 1. And the area under the graf is infinite.

[@artunsaday6391]

@@Antediluvian137 But that is not the graph of all possible outcomes. It is just a function that tells us the probability of a certain outcome (48 out of 50 in this case) as a function of the actual succes rate s. This is why the whole curve decreases as the data increases. Since there are more possible outcomes in the actual pdf, that certain outcome has a lower chance. As an example think about the similar outcomes: 478/500, 479/500, 481/500... they all have similar probabilities to 480/500 so they are less likely than getting 48/50. At least this is what I understood hope it helps.

[@pontust9773]

Dude you gotta release the other parts soon, I got an exam in statistics coming up😂👌

[@sajanator3]

How did it go ?

[@pontust9773]

@@sajanator3 I BARELY passed but hey, I passed ;)

[@sajanator3]

@@pontust9773 Good job :)

[@eccentricOrange]

Lol I'm in the same boat. Exam soon

[@sajanator3]

@@eccentricOrange I've got a prelim soon so I'm working my ass off. Lol

[@GuppyPal]

The quality of your videos is off the charts, man. Color coding things in the formula was SO smart and helpful. Really great stuff here.

[@PracticalEngineeringChannel]

Love thinking about binomial distributions. A lot of what I do at work is looking at flood magnitudes and their probability. A lot of the way the constructed environment looks (and costs) is based on societal risk tolerance and where we choose to draw those lines in the sand. Great video!

[@noob78]

How do you calculate the probability of a flood of magnitude of x?

[@-guitarhero]

@@noob78 they're engineer scum. Probably using 'proof by python simulation'.

[@boiimcfacto2364]

Me who literally just finished Baye's Theorem, seeing this at midnight: *So this is the power of Ultra Instinct?*

[@solarisone1082]

Found the Dragon Ball fan. :-)

[@SwainixFPV]

This video just reminds me I'm already rusty on the bayesian stuff I studied at the start of the year lol

[@tanta1519]

*Requiem

[@nameunknown007]

No, this is Google spying on you.

[@deus_ex_machina_]

@@nameunknown007 But the video was published on the day of their comment. If it was released earlier and recommended today than it could be evidence of Google spying.

[@DrBlo0dy]

If i can go back in time re-start college as a freshman, I would binge watch all of his videos. He makes math so much easier and interesting to understand.

[@undeadman7676]

I had this exact same question a few years back and couldn't figure out for the life of me how to quantify my instincts. This video has saved me ;--;

[@festusmaximus4111]

I'm so sad that I've outgrown the point of these videos in teaching mathematics, I can't express how useful the essence of linear algebra and essence of calculus were in building my first intuitions about those topics, but now that I'm well on my way towards a physics masters there's less new maths here, less in total to learn. The reason I still watch them has therefore changed; now I watch not to learn the subject, but to learn how to teach the subject. In this regard, the ways of teaching presented in 3B1B's videos have been invaluable to me, as I have always loved to teach. So thank you, because while the total amount I can take from each new video is now diminished (through no reduction in the quality of the videos), I may now take different things from the videos, and instead of marveling at the beautifully precise animations and newfound mathematical understanding I can wonder at the phrasing of each complex topic, and how even the most complex ideas become simple when you speak.

[@quadrannilator]

There are many here who may never teach, but watch for the precise reason you do. Its the approach, not exactly that we do understand the topic as adults with a little bit of brain and practice. Happy Teaching!

[@unsweeteneddoll]

Are you a unicorn? Man, it feels like every math teacher I’ve had has only taught the subject bitterly and with resentment towards both the subject and their students. Of course it’s not true of all math teachers, but unlike language arts, it feels like math teachers who are passionate about teaching are rare. Good on you.

[@abhishekthorat3631]

Your Ted talk was awesome just loved it.

[@yahav897]

He's done a ted talk? Dang, I should watch it.

[@coffeedude]

Is this a meme? If not how can I find it?

[@dakshitchopra04]

@Walter White I was in middle of it when this notification appeared .

[@soumyadeepghosh92]

@@coffeedude kzbin.info/www/bejne/qZCvXpmmbcx3sLs

[@soumyadeepghosh92]

Thanks for the info . Cons of underestimating TedX , and not hitting the notification bell .

[@elvinchateauvert]

Since all my classes have moved to online, this is one of the best supplemental "classes" I've ever had :)

[@tobiasopsahl6163]

Please continue this series :) I know you have a lot to do, but I am really looking forward to the rest of the probability series.

[@rogerwang21]

I’ve been wondering about this topic for a long time now. Thanks for a video on it!

[@mattoita]

The beauty of this channel is that, even though I might be well versed with all of its content concept-wise, I still enjoy very much watching these videos. I'm just delighted to see how you choose to present concepts, the graphical effort to show them clearly, in coordination with speech. Sometimes in incredibly revelaing ways. This is what teaching is truly about, and it is kinda sad how often teachers themselves mistake their job as "giving concepts to students" (...and testing them) (or something along those lines). Your work is very inspirational, instructive, enjoyable. I'm so happy that I know you - and big props to each one collaborating to make these videos. I'm confident in saying that you're like friends for many of us, even though we don't know you in person!

[@nitishkumarthakur1920]

and soothing voice of presenter

[@MathPhysicsEngineering]

For those who are interested in detailed analysis and proof of the binomial expansion formula, I would recommend looking up on my channel the video called:" The Binomial Expansion Formula Derivation and Proof"

[@unknowninfinium4353]

I am a noob. But lately my interest in statistics is growing. I am also not that smart and ny IQ is below average. Could you suggest me some books or so about statistics which I can start reading to understand these concepts?

[@skydivenext]

giving concepts to students is a good way to study you dont need these videos to understand them, the problem is that teachers regurgitate concepts with no much explanation reading powerpoint slides when they dont even rember much about the concept either

[@stxllr4687]

Can’t believe watching a Minecraft KZbinr led me to this point. I can barely understand anything but it’s kinda interesting now

[@D00000T]

well if you’re still in school and haven’t taken a stats class yet, it’s almost inevitable that you’ll have to slog through this at some point

[@xericicity]

@@D00000TAnd then immediately forget everything the second you leave the exam hall and rediscover it all again, after 10+ years of working as an engineer, because of some weirdo projects that needs it.

[@levinunemaker9406]

I don't know what insight this offers, but seeing you add the probability of having a good or bad experience as a superposition of states made me think of that small change as an infinitesimal, as in differentiation, of the probability measure. I was delighted to find you have a whole series leading to the beta. Great work! Thank you for all your help.

[@Vertic03]

A 3blue1brown has 1617 positive ratings out of 1621 reviews. What is the chance of the video giving you a positive experience? Easy, it's 100%

[@NacToYT]

But that's not dependent on the ratings. That's a simple fact.

[@VidheyOza]

r/UnexpectedlyWholesome

[@sebastianjost]

The others just forgot to leave a like and those who disliked just misclicked. ^^ I totally agree with your comment.

[@ShannonJacobs0]

So you provoked me to give it a thumb's down (but the deeper reasons are in my earlier comment).

[@acey457]

i will thumbs down but tomorrow i will wash my hands of it

[@gabrieldiazireland7157]

Thank you for all your videos and knowledge explained in such a good way. I am sure it helps the world to be a better place with all the all the engineers, mathematicians and physicists using them.

[@teachmemaster2669]

You sir are a major MVP! I don't know what I would do without this video. So thankful for all the knowledge and the lucid explanations/visualisations! Keep up the good work!

[@weitt14]

Great work once again! I would love a whole series on an intuition for statistics just as for linear algebra and calculus. They really changed how i view my world

[@zetsui0411]

4:56 probability of probabilities 5:45 in context, proability of data given an assumed succes rt 8:35 binominal 9:10 animation

[@kazimraza6278]

Brilliant. This is exactly what we encounter in the plant safety reliability as well. Probability of (equipment fault|safety function fail). Excited fpr the next episode.

[@DennisZIyanChen]

you are so incredible. There are a lot of cool statistics concepts and I can't wait to see more videos on this from you. Just so inspiring to see how well you demonstrate these concepts.

[@ow4744]

Can I just say that as someone currently studying statistics this video proved incredibly interesting and useful - suddenly when we got to about 10:30 I realised we were essentially talking about the same maths that is behind confidence intervals. Very helpful to see it in a different context!

[@marco_gallone]

Hi Grant! I loved this but I’ve been on a 9 month cliffhanger! Please release part 2 and 3!

[@yordan3146]

Hi 3blue1brown, we are still waiting for the two other videos announced in here, "Bayesian updating. Probability density function" and "Beta distribution". Are you thinking of publishing them soon? Thanks in advance for your help to understand probability and have an intuition about it. It is really something. Great job!!

[@wealthy_concept1313]

Yo please which course is this at college?

[@fatfr0g570]

@@wealthy_concept1313 Don’t take my word as advice, but I’d assume that this (binomial distribution) would be found in introductory statistics. As to the others, I am unsure as I haven’t heard of them. If you had asked this at the end of April or very beginning of May, I could have asked my Calc 2 professor since he also teaches statistics.

[@wealthy_concept1313]

@@fatfr0g570 ok

[@andresperezrobinson2247]

Still waiting

[@oscarespinosa1894]

Still waiting

[@queenquit]

Thank you for this video! I'm a first year master's student in biostatistics with no math/stat background. After first two weeks of study I feel completely lost, but luckily your amazing videos come into help! They saved me!. I really love them and hope you could have more video on statistical inference. A lot of people would love them since many are interested in stats and machine learning and are looking for jobs in data science. Lots of thanks again!

[@Israelpwn]

Oh, what a blessing you and your timing are, 3B1B! My work mainly involves improving a classifier that my company made, usually approaching it from the data view of things. However, I've recently started a book on Bayesian statistics and was curious to approach the problem as a Bayesian probability instead of raw data. Needless to say, I'm super psyched for this series!

[@notsoclearsky]

You're making this so easy while in the book it seems like a nightmare. The book makes me lose interest while this video reminds me why I always wanted to take and eventually took maths and science in the first place.

[@Thmyris]

The good teacher syndrome. Maybe try reading another book on the subject? After all, books are inanimate teachers.

[@pranavlimaye]

I absolutely agree. I too had to ditch the textbook and teach MYSELF these concepts. Sure it was fun, but probably the most exhausting thing I've done. Ohh how I wish Grant had uploaded this one before my exams came up

[@alveolate]

omg if only i had this video when i first had to do binomial distribution for A level Maths C... the curse of Gen X'ers being born too early to enjoy youtube explanation videos before key exams.

[@sankarsanbhattacheryya7498]

I'd suggest start doing some solved examples to get the hang of it. Read each line, try to guess what you should do for that line and then verify whether your approach was true from book. My introduction to statistics class was also a bit hairy but i found doing the solved examples really helpful

[@pranavlimaye]

@@sankarsanbhattacheryya7498 are you Indian? Which university/institute did you join for a statistics class?

[@davideizzo2683]

I learned that damn binomial distribution like 4 times in my life now, hope I'm not gonna forget it this time

[@walkedoutt7225]

Thank you so much 3blue1brown you’ve inspired me to double major in math and economics. You’re videos are absurdly intuitive. So much so that even layman can understand. This is true intelligence. Keep up this great work buddy.

[@dancinindadark]

Learning cumulative binomial distributions just to prove how absurdly lucky my friend was and unlucky my other friend was with drop rates in a game has been very fun and educational thanks to your short series, thank you

[@shkotariq6138]

clicked faster than a someone with corona deciding to have a vacation all of a sudden

[@peterbrough2461]

You forgot to subtract five ✔'s for reviews by the seller and his friends, and two ❌'s from the seller's competition - right off the top.😮

[@letao12]

Yeah, the underlying assumptions are really important. A completely random model with independent ratings is good for teaching math, but doesn't really match reality.

[@shrikedecil]

Not to mention Amazon shadowbanning, deleting, and otherwise monkeying with the numbers. Garbage in, Garbage out. Now, for advanced statistics, we'll play with poll results....

[@SumNutOnU2b]

You beat me to it. I was about to comment a question asking if they plan to cover this sort of thing

[@andrewdunbar828]

What's the probability that a seller has only five friends and two competitors?

[@doombybbr]

I would have instead have increased the number of artificially generated data, to 10 artificial numbers instead of just 2. Generally put the number at which you would stop being suspicious of your data sample as the amount you inject. Doing this actually causes option 3 to become preferable. Though it also means 1 review sellers with negative get screwed over, so your method is probably better.

[@klumpeet]

Thank you for this excellent start to this series. Explaining the concept of a likelihood function this clearly, without all the awful jargon, is admirable. The difference between data|s and s|data is too often misunderstood in my scientific community right now. Thank you.

[@WhitefoxSpace]

3blue1brown is without a doubt one of the best ways you can start your day. It gets your mind working. One of the rare creators I can watch and not feel like I'm procrastinating.

[@anch95]

5:20 "Nikola" 😂😂 I see what you did there.

[@snuffeldjuret]

it actually exist :) en.wikipedia.org/wiki/Nikola_Motor_Company

[@taratron]

what's the joke?

[@nao10270]

@@taratron car factory named "Nikola" is referring to Tesla the actual car compagny, which in turn is named after The physicist who discovered AC : Nikola Tesla

[@Eurotool]

3B1B knew NKLA was a fraudulent car company so he used it as a fictive example

[@faresalouf]

@@Eurotool finally someone who understood a thing or two here

[@PsychoSavager289]

"Ships from Plato's cave" I see what you did there.

[@threehotdogs]

your videos are just really visually appealing, i don't personally find math very engaging usually but your video production and presentation are really top notch

[@autishd]

This is a fantastic video. Anxiously awaiting parts 2 and 3.

[@sergniko]

Очень интересно и понятно. Теория вероятности после таких видео становится простой и очевидной. Спасибо за хорошее видео. :-)

[@NOVAsteamed]

This channel is so amazing like seriously. You made me decide to choose a math major with computer science.

[@samoldfield9997]

I have never enjoyed a probability course this much. these visualizations are really amazing.

[@rishiagarwal926]

Probability distribution was a thing that I was taught recently in High school, but I never got what each term ment or why Binomial theorem was involved in it, and now I even understand it's graphs! Thanks so much!!

[@crh231]

"Try using a Non-Gregorian calendar" made me chuckle

[@NoriMori1992]

For February *31* 😂

[@juliocardenas4485]

Was the third video in the series ever posted? I’m dying to watch it

[@asdfvp5795]

Mr. BlueBrown, you have the magical gift of making incredibly difficult sounding mathematical concepts explain in a way, so I can understand them in less than 10 min. Moreover, I find it super logical when you do so and I can reproduce this by heart. No math, economics or accounting teacher has been able to do this in my 5 years of uni.

[@maibrl5251]

I just want to thank you for the series, that’s exactly what I need to understand for my upcoming maths finals, at least part 1 and two afaik

[@nickcampbell3812]

I was looking for part 2 not noticing this came out an hour ago. I'm used to watching older material and just moving on to the next. I will wait.

[@FlockOfHawks]

big grin : same here 👍

[@bencrossley647]

I came up with a “solution” to this about 3 weeks ago. Makes me happy to see a formal way to do it :p

[@dakshitchopra04]

more reviews means more genunity of rating . That's why I prefer more reviews , when two sellers are selling same stuff .

[@bencrossley647]

My “solution” was: average + 0.1log_2(votes/32) with a cap on this term at votes = 128. The thinking being that if enough people have voted for something that in itself contains value (limited to +0.2)

[@Nothen]

@@dakshitchopra04 Only issue is that some sites like ebay & amazon have problems with vendors inflating their review numbers artificially. How much it affects the number of reviews, I have no idea.

[@DanielGonzalezL]

The real solution is seeing which seller is closer to you and can save you more time :P

[@bencrossley647]

Daniel Gonzalez You mean the Traveling Amazon Salesmen problem.

[@Silver-ue7ev]

I love the way you use animated graphs, it really helps the video be more understandable

[@tigerbelly2152]

Hi Sir, I love your series. Very concise and clear explanation delivered masterfully with visuals. Thank you! And please do more on prob and stats theories!!!

[@bigwheel9468]

Dreams ender pearls be like

[@princeo242]

Lmao 5Heads coming in

[@D00000T]

these children should just watch videos like this, learn stats themselves, and then decide who’s right themselves. They’re most likely going to have to slog through a stats class eventually so why not start the pain early when it won’t effect your life greatly if you fail?

[@leoofficial527]

Lol

[@rmanami]

what

[@bradysmith6339]

0:12 Ah yes, I would like it delivered February 31st

[@pergrinepi3130]

Bruh

[@BootesVoidPointer]

lol, what

[@cansabanci]

also i would like it to be shipped from plato's cave and interdimensional shipping.

[@PranavSNair-hc9zk]

seems unlikely

[@mahimmarufuzzaman3588]

LOL

[@mohaghaderi6077]

Thanks for such elegant explanation of a deep and extremely important topic in empirical modeling. Great video!

[@anasshaikhany9733]

Please continue this series it's been 4 months from now, I am dying for some lovely probability math, Love from Syria

[@johnchessant3012]

"All Prices Pi Publishing" is selling the $75 book for $314.15. Go figure.

[@tiny_toilet]

I'm tempted to give a dislike and ask my partner to give a like for a more accurate rating of this video.

[@Ruhrpottpatriot]

I'm tempted to do that about your comment. ;)

[@appleoftheground]

big brain

[@rabbitclimbmountain]

Best comment of the day.

[@orange1304]

@@Ruhrpottpatriot It wouldn't work though, since we don't see the dislikes, sadly...

[@tiny_toilet]

@Hugo Vendetta Wasn't me. I could NEVER actually bring myself to give 3Blue1Brown a dislike!

[@muzamilhussain7850]

In love with this Channel, Hats off! you are not just making the videos but serving the society in a better way

[@simplemathematics_25]

Best Math English Channel. I wish more teachers had your abilities to teach.

[@icicleditor]

This is all assuming that the buyers aren’t paid to give positive reviews. *COUGH* RAID SHADOW LEGENDS *COUGH*

[@Raveyboi]

Bro, do you have corona

[@icicleditor]

nah, it's just a cough for comedic effect (sniffle)

[@khytron06]

@@Raveyboi When i read back the comment after reading your comment i was so dead, your comment deserves more likes.

[@Manly-Tears]

RAIDDDDDDDDDDDDD SHADOOOW REGENDS: Only gained attention because of the Internet Historian Vids, and initial paid reviews.

[@kemchobhenchod]

Using review filters (ie. reviewmeta and fakespot) and reading their trusted positive/negative reviews first to avoid confirmation bias/cognitive dissonance are also very effective.

[@dasdaleberger5683]

"Ah, fortune smiles. Another day of wine and roses, or in your case, beer and pizza!"

[@davidm.johnston8994]

Great video Grant, as per usual on your channel! Thanks for all the work that you put into these. I think you're truly at the cutting edge of explaining math with these videos. Take care ;-)

[@bryanroland8649]

Lucid and enlightening as always, even for innumerate viewers like me. This video helps you to get a grip on the counter-intuitive nature of probabilities.

[@jandew314]

1:52 When handling a 5-star rating system, like how Amazon gives you the distribution of those ratings, can you do a similar thing of just adding one of each possible rating (one 1 star, one 2 star, ... one 5 star) and then compute the expectation value of the resulting distribution?

[@yuxin7440]

I think I will deal with this problem like this: We assume each seller as a Bernoulli random variable with probability p of getting a good review, and each review is *independent* (which is not the case in reality). Given a sequence of n reviews X_1, ..., X_n, we can define the *likelihood* of this sequence given p as L(X_1, X_2, ..., X_n | p) = \prod_{i=1}^n p(X_i | p). The specific value of p that maximizes function L, denoted as p^ (p-hat), is the MLE (aka. Maximum Likelihood Estimator) of p. We can compute MLE using calculus techniques (derivative of L = 0 and second derivative of L < 0). Of course, MLE is not always the best estimator, we can determine whether an estimator is *biased* by testing whether E[estimator] = actual parameter. It is not biased if they are equal, and vice versa. To get a more practical view of the estimation, we can construct the *confidence interval* of the estimation. By central limit theorem, the distribution of the sum of n independent identically distributed random variables is normal as n approaches infinity. If the estimator can be represented as some constant times the sum of the X_1 to X_n, we can construct the x% confidence interval (we usually use x=90, 95, and 99 in reality) by P(actual is within the interval: [E[estimator] - z*stddev(estimator), E[estimator] + z*stddev(estimator)]) = 2\Phi(z) - 1 > x%. Using the standard normal table, we can compute z, and get the interval.

[@Wonders_of_Reality]

I love this topic! Grant, many thanks for finally showing us the binomial distribution. Oh, so many memories! I even wrote a flash-script that was calculating it.

[@ditrizna]

Please, please, please continue these series!

[@johnchessant3012]

3:08 the facial expressions on the buyer pi haha

[@tomf3150]

Long story short : Read the negative comments.

[@kiattim2100]

I used to do that with books and ended up not buying anything.

[@chrisjfox8715]

Only thing, there’s almost always a collection of asshats in the comments claiming some product/business is the worst thing ever and that you shouldn’t buy it. The specific criticism is what to pay attention to and whether that specific thing is something that matters to you. Because chances, there’s always a few people that happen to have a negative experience and assume that literally everyone else will experience the same.

[@whirlwind872]

@@kiattim2100 Agreed, reading the negative reviews of movies/games/books will poison your mind and convince you that it's bad before you even give it a fair chance. That's the problem with the internet, you are so often told how to think rather than coming to a conclusion entirely on your own

[@mazengwe28]

What I do is filter out the ratings. If it's a 10 point/star rating, just go by the rating bewtwen 8-3. It tends to give the most honest results because it has room for error. 9's and 10's say that everything is perfect (which is hardly the case) and 2,1, or 0 tend to be too negative and just bad mouth it without giving any objectivity.

[@BenjaminCronce]

10% of people complaining that it breaks after 6 months.... no thanks.

[@an1ketsharma]

Was Studying (still studying) probability when this poped up in recommendation. Worth the watch ngl.

[@unoriginalusernameno999]

Nice video and FANTASTIC TED talk by the way! Congratulations! Keep up the great work!

[@binarywizard69420]

Time to reck dream with phd of binomial distribution 😎

[@sword013]

Please make a whole probability and combinatorics intuitional playlist in deep !!!!!

[@TheVixenfan]

I hope the next one comes out soon! love your videos for supplemental learning material!

[@CyberWorldwithPratik]

the way he explains maths... it's just amazing. I really want more and more videos like these.

[@andreykey2623]

Where is the beta distribution video🙈?

[@billywongting]

An actuary who is tired of calculating give you another solution: - Surely the one with 200 reviews is better. They get more customers even offering the same price!

[@LddStyx]

Which is more likely that all the customers are rational actors and have done the math OR that they used your heuristic trusting that the more popular one is better? :p

[@absqrb]

@@LddStyx they would all start w/ 0 of 0 though

[@MegaDardery]

@@LddStyx snowball effect

[@samsibbens8164]

They're not necessarely better but if the percentage rating is one that if were 100% accurate would be worth the risk to you, then the one with more reviews is better because at least you're much closer to the actual risk you're taking when buying from them.

[@shadsluiter]

Of course. When choosing a restaurant, would you dare enter a place that is nearly empty while the surrounding places are full?

[@Squash101]

I wish this series was continued, A video on poison distribution, geometric and a few others would be amazing to watch!

[@nikmat]

I’ve been taught couple of times what bayes rule in my college life, couldn’t grasp it till now, I’ll be giving another go at your part2

[@ritiksinha5651]

When you are in the Faculty of medicine. you: Just need to review it for the genetics lecture.

[@Jessie_Helms]

Personally in this type of situation I look at the stars. If there’s a LOT of 1 and 2 stars with almost no 3 or 4 stars, I take that as a bad sign. If there’s a lot of 3 or 4 stars that’s better. If it’s mostly 4 or 5 stars with a handful of lower ones that’s really good.

[@ceruchi2084]

I trust 4-star reviews more than 5-star, because the 5-star ones are often friends or paid accomplices.

[@noahlewis7443]

I love that he starts with a real world application of the material in the video!

[@scott907]

Really enjoyed this, used this at work to risk a project. Glad to hear my thinking was along the right lines

[@adityachk2002]

The last time I was so early I was still learning essence of calc

[@JTheSpearman]

Where is part 3?

[@Alkis05]

I always wondered about this, but never gave it proper thought. That is a very down to earth and useful in everyday life piece of statistics. Thanks, man.

[@hamilpatel4025]

cant wait for part 2. really interesting stuff!