Bayes' Theorem - Example: A disjoint union

In this video we work through a Bayes's Theorem example where the sample space is divided into two disjoint regions, and how to apply Bayes' Theorem in such a situation. This example of a "two bucket" Bayes' theorem problem can be extended to 3 or more buckets. Most "real world" examples don't involve literal buckets like this problem, but a division into two categories such as "vaccinated" vs "unvaccinated".
The formula: P(B1|A) = P(A|B1)P(B1)/(P(A|B1)P(B1) + P(A|B2)P(B2)
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[@quitekid2863]

Crazy how he explains it the way I want it to be explained. That’s an ideal teacher right there.

[@jerica2432]

You taught me this for free much quicker than what my college tuition-paid professor ever could.

[@niloykarmaker9106]

same tbh

[@beastworld8109]

amen lmao

[@musancube2308]

I ve failed statistics three times but through ur tutorials I ve regained my hopes in passing thi module. thank you

[@kashishhashim6471]

You got this girl 💪❤️

[@Lana-jg7lg]

Musa Ncube I hope you passed bro

[@sinnerg8968]

I have given up my statistics major and almost shifted to arts and commerce but after I saw your 8 minute and 31 seconds of pure awesomeness, I have decided to stick to my major and be a failing student.

[@okey5]

I tried understanding this for ages from my textbook unsuccessfully, this helped immensely

[@francys5567]

I am so glad I've found this video. The Bayes' theorem is made so easy, thanks very much, great job!

[@Statemachine709]

This was incredible. Your excitement for the problem helped make the theory exceptionally clear and concise. Thank you!

[@melchiortod29]

I don't think you know what a daddy you are. You help us uni guys so much, thank you so much!

[@suyashsreekumar3031]

Thank you sir for your clear and simple explanation of Bayes theorem.

[@raghunandanbs2005]

ur such an amazing teacher what i failed to understand in my class after listening to an hour of lecture you thought the same thing in a better way and that too in few minutes . thank you sir

[@Wayfarer17683]

Years later, thank you sir. This helped me so much. Very detailed!

[@mehrishyousuf592]

Thanks for this simple yet clear explanation.

[@sdsa007]

Bayes theorem was a puzzle for me for manymany years, because my education was frequentist, and the stats people were still disputing each other. My bosses boss once called me into the office and asked me about the Monty Hall riddle.... I just happened to guess the right answer! So I got to keep my job! Anyway, 'I like how you said 'having balls..'.. all mathematicians must have balls, including the women because it automatically means you have an infinite number spheres and circles too! Beautiful harmonic shapes, and now I understand something that is fundamental to being a scientist and a mathematician! Thank you!

[@ayushsatyam2616]

Great work sir the video helped me so much. your idea is a bit different and simple from the commonly taught expression of Baye's theorem.thanks

[@bineetapareek5316]

i love you, i have literally never understood anything in life better

[@taqiyasir9216]

Thank you Sir for the beautiful explanation.

[@sahanaparasuram1948]

I literally watched just this one video and now I can solve soooo many problems!! Thanks a lot 😄

[@DrTrefor]

Glad it helped!

[@jasmineakuacaut-pinto2924]

I really enjoy ypur teaching. Makes maths so simple to understand. Glad i found you 🤩🤩

[@skyboundzoo7529]

Awesome teaching - thanks for the knowledge

[@harrychecker5509]

Beautifully explained.

[@disabledassfk]

You deserve more recognition 🙇

[@riddhispeaks9662]

just love your videos, understood why-to-do instead of the traditional what-to-do, ad I just wanted to keep listening, which is not common for me, thanks

[@patrickbeard3786]

I also calculated 5/12 for P(A) by simply counting up all the balls (12) and all the blue balls (5) and voilá 5/12. It's nice to know they match.

[@gabrielamartendal1466]

Você explica muito bem, parabéns.

[@yoseppandji8698]

Thank you, man! You help me a lot hahaha. Continue the good work!

[@observever7808]

Brilliant example!

[@okaythisisepic3212]

Great video, clears up a few things for me :) Just a minor thing you could improve would be to write your equals sign a tad bigger since they now can be mistaken for just a 'times' dot, I guess it's obvious from context but I jumped into one of the videos and wasn't completely sure about the equation first

[@streetbeats1672]

very well explained ....Man i wish I had your logic!

[@amjad6361]

Great job sir, may God bless you!

[@yousufazad6914]

Awesome explanation!

[@michaelbauers8800]

The people who downvoted this got here accidently while searching for Beyonce's latest video

[@saurabhbcn]

Michael Bauers hahahah

[@mgmartin51]

I was looking for Beyoncé’s theorem too.

[@epiren]

True Story

[@rahul2k208]

Thanks sir wish i had a math sur like u

[@skygautam]

Amazing explanation thanks .

[@dexterns7369]

That was really helpful, thank you very much

[@haidernaqvi545]

Very Nice way of explanation

[@baishalipalodhi]

Good Explanation!

[@Manofthehours]

I have learned more in 1 day than in a whole semester

[@jesannafew8590]

Thank You so much for helping out with this.

[@adamTDE]

Amazing pedagogy!

[@kanha12297]

Best explanation ❣️

[@tanwenxuan2999]

Thank you ,this is very useful to me.

[@zhenminliu]

Simply explained. Thanks

[@azbenzaki5542]

you're a hero man!!!

[@josephcold]

This is so so so good.

[@viraj_singh]

that helped me. Thanks a lot.

[@pratikprajapati9133]

Nice explanation thank you

[@simphiwesono5635]

Very helpful, thank you

[@MegaFanFan21]

You're AMAZING !!

[@irenemutesi7045]

Thank you so much atleast am gonna pass college finally

[@Mega_Mikey]

I appreciate the breakdown of the first part, but can’t we just look at the two buckets and go “ok there are 12 balls total and 5 are blue...5/12 is the answer”?

[@TheGuywithnolife]

exactly what im thinking

[@earlworth]

Here you can count them, but in many problems it is the denominator of Bayes' theorem - the "given something else happened" bit - which is the hardest thing to calculate. Here's another example: P(criminal is guilty | evidence) = P(evidence | guilty) x P(guilty) / P(evidence) but what is P(evidence)? It is P(evidence and guilty) + P(evidence and not guilty) - or in other words: P(evidence | guilty) x P(guilty) + P(evidence | not guilty) x P(not guilty) - and that is equivalent to the video: P(A | B1) x P(B1) + P(A | B2) x B2 . The whole point is that counting the balls works only when you can indeed count the balls, whereas the process he showed works in a far broader array of situations :)

[@EvTrev11]

in this example yes, but the data types in the separate buckets could be different and may not lend themselves to being lumped together, so calculating the probabilities separately is then useful

[@clickherenowz]

So helpful thankyou!! Could you make a playlist for all of your Stats videos?

[@topogigio10]

Fenómeno Trefor, besos de Riquelme!

[@sanobet882]

Its so hard method even the balls thing is really so hard but you put alot of effort in to it so thank you ❤️❤️

[@vaibhavsingh1049]

Thank you. This helped

[@lanaf3067]

Thank you this was so helpful!!!!!!!!!

[@dbb1611]

good stuff. i love it

[@andrewharrison8436]

Bayes theorem - underappreciated - probably a bit too new for many people (published after the Rev Bayes' death, he died in 1761). Nice explanation - thanks

[@thesclasswithsilas]

Very helpful👏

[@shakirinjahanmozumder2572]

Thank you Infinity times....

[@madridi891]

that's pretty awesome thank you very much

[@kareng4120]

Can u make more videos this is really helpful

[@mukulnimrani1062]

what is 1-3/5 i.e. 2/5? is it the probability of drawing a blue ball such that it is not from the first bucket?

[@nexusbeing6403]

It was nice sir....you solved the problem at the point where finding P(A) and differentiating it with P(A|B) was concerned.... Thank you sir..... But please tell me what do you call that stylus thing using which you write this ??

[@alteshaus3149]

Thank you!

[@ojsawe42d16]

You taught me this in 10min ,while my teacher couldn’t in 5 months

[@Loots1]

Thanks for having the courage to talk about your blue balls!

[@yusmanisleidissotolongo4433]

Thanks so much!

[@ronarprefect7709]

The way I did it mentally was sort of compute the blueness of each bucket, i.e. 1/2 for the first and 1/3 for the second. I then added the two bluenesses together to get 5/6. I then computed how much of the blueness was contributed by B1 by dividing the blueness of B1 by the total blueness(1/2 divided by 5/6 = 3/5). This should work for any number of buckets.

[@CodehanCodes]

Thank yous so much sir

[@suhanikataria1819]

It would be bit easy for me to understand with subtitles. But you explain good😊

[@PMayhew]

Has the equation P(A) = P(A ∪ B1) +(A ∪ B2) been derived from the probability of union of two events rule P(X∪Y) = P(X)+P(Y)?

[@karannchew2534]

Didn't get the P(A) = P(A∩B1) + P(A∩B2) bit. Probability of A (getting a blue ball) is the probability of A and B1 (blue ball from bucket 1) plus probability of A and B2 (blue ball from bucket2). Why the "intersect" ∩ ?

[@aizadanayal8727]

Would the probability of finding a blue ball from bucket 1 or bucket 2, be called an independent event or conditionally independent event?

[@aaronisucama4390]

Thank you

[@singcovers8479]

thank you so much

[@rajkumar-ub7wc]

Thanks Sir

[@shaunaweisser8575]

THANK YOU !!!!

[@suhailmusa4450]

Suhail's Theorem to get P(A): Count Blue = 5 Count Yellow = 7 Sum = 12 P(A) = 5/12 Nobel Prize here I come

[@somadutta9962]

bruh

[@fplasil]

Thanks for the vid. I have a question. For the calculation of P(A), can I just think of it as having 5 blue balls and 7 yellow balls, together 12 balls. So probability of drawing a blue ball would be 5 out of 12. Is it a coincidence that the P(A) comes out 5/12 using this simplification, or would this idea be still valid logic even though we have two separate disjoined buckets?

[@andrewharrison8436]

Because there are the same number of balls in each bucket your logic works. However suppose that the second bucket had twice as many of each then the method in the video works but aggregating the balls would give the wrong answer as each ball in the bucket now has a smaller chance of being picked, (1/2 for the bucket being picked and then 1/12 for each ball so 1/24 each).

[@gooddeedsleadto7499]

Could u please show with an example how can we use Bayes theorem to predict who is more likely to win the presidential bid? Thanks

[@rickyzheng9018]

you saved me :)

[@chintandd]

thanks. in 7:00 timeline, How P(B1 | A) comes? since A is unknown to us. Shouldn't it be P (A | B1),where B1 is know and A is unknown?

[@ferndaily5377]

I lol'd everytime he said blueballs

[@ajaybanstola123]

One question, trefor. If the essence of Bayes theorem is to update the prior belief, what do we update here?

[@ajaybanstola123]

oh okay. so if we repeat the experiment, we might get to a constant value of the probability then?

[@harouttatarian3255]

5 greens balls in total 3 greens in B1 and 2 greens in B2 P of B1 given that the ball is green is = 3/5

[@Neale91]

Similarly, since both buckets are equally likely, P(A) is total blue balls out of total balls which is 5/12

[@user-bu8mg7uq3s]

thank you

[@ajaybanstola123]

at 3:53 why did we add the intersection? How does (a intersection b) differ from (a|b) in intuitive sense.

[@ajaybanstola123]

​@@DrTrefor I did realize it later on. Thanks for clarifying.

[@10cgokulramh70]

I tried solving case for p(blue ball/B1) and got 6/5 is that correct ??

[@MsUlmer]

Question: when moving P(B) by multiplication... how does that change the relationship from an intersect to a given that? Example: P(A|B) = P(A n B)/ P(B) Multiply the denominator by P(B) = 1 Then the numerator by P(B) = it becomes a given that | rather than remain and n. OR P(A|B)*P(B).... How/why does it go from an intersect to a given when the denominator is multiplied to the numberator?

[@MsUlmer]

Eureka. I see where my thinking broke. Forgotten algebraic rule. You must do the act on both sides of the =. Given that P(A|B) = P(A n B)/ P(B) AND I multiply up P(B) THEN I must do the multiplication on both sides of the equation... Resulting in: P(A|B) *P(B) = P(A n B) Right??

[@blAnk7300]

Excuse my ignorance but I'm having a hard time understanding the difference between P(a|b) and P(a inter b) in real life. You said that P(a|b) is the chance of a given b but you also said the same thing about P(a inter b). What am I missing here? (Beside some IQ points)

[@blAnk7300]

Thank you for taking the time and effort to reply. Subscribed !

[@vonderasche2963]

Why did you move P(B1) and P(B2) from the denominator to the numerator

[@tygriffin5528]

I'm enamored with Bayes theorem. I watch the videos and understand them, but I don't find myself thinking intuitively using Bayes or conditional probability. So I want to know if thats ever a thing. Do the people who study this a lot, or teach it, ever reach a point where they see it all around them? I enjoy it either way but I have wondered whether I'm just not wired see and use Bayes creatively in life.

[@cassandraweaver8479]

I think I'm your biggest fan! lol

[@mazharmumbaiwala7451]

Stan, is that you?

[@Madhu-wr4om]

hey, your video was really helpful! If there were three different variables to consider like what if we have three different coloured balls, will the same apply?

[@DrTrefor]

Absolutely, the formula generalizes to n variables nicely.

[@Madhu-wr4om]

@@DrTrefor Okay thanks!

[@hopesibeko8059]

thank u

[@zeinelmokhtar4105]

You're great! You just need a better mic 🎤 and that's it 😉

[@mariamalavi2679]

God bless you. Thats all i have to say.

[@RMD551]

Very Epic

[@civenge1802]

Knowing there are 2 buckets, one with 3 blue balls and one with 2 blue balls, my initial thought was I don't care how many yellow balls there are. Therefore, of a total of 5 blue balls, 3/5 chance of being in bucket 1. I'm not sure if this is flawed logic, but it made more sense given the initial question. The actual math is important for more complicated questions.

[@willjohnston2959]

Your logic works if number of balls in each bucket is the same, but breaks down if that's not the case. Consider if instead bucket 2 had 200 blue balls and 400 nonblue balls. Bayes theorem still gives 3/5 as answer. It doesn't give 3/203 as answer.