Intro to Bayes’s Theorem | Probability Theory

What is Bayes’s theorem for conditional probability? We'll be proving it briefly in today's lesson, as it is a direct result of the definition of conditional probabilities. We'll also see how Baye's theorem works with a sample space partitioned into separate events, and an example of applying Baye's theorem! Baye's theorem gives us a way to calculate the conditional probability P(A|B) using P(B|A).
We know that the probability of A given B, P(A|B), is equal to P(A and B)/P(B) and P(B|A) = P(B and A)/P(A) = P(A and B)/P(A). Then, multiplying both sides of the second equation by P(A) we have that P(B|A)*P(A) = P(A and B) and we can substitute this into our expression for P(A|B). Thus, we have P(A|B) = P(B|A)P(A)/P(B), and this is Baye's theorem (sometimes called Baye's rule). Often times, we will have to calculate P(B) in the denominator using the Law of Total Probability.
I meant to put this in the video as a caption: the example problem is from "Probability" by Jim Pitman.
Intro to Conditional Probability: • Intro to Conditional P...
Law of Total Probability: • The Law of Total Proba...
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[@_Ayush_S]

No-nonsense, completely on-point, and no focus on showing his own video and wasting time with useless information. Thanks a lot for explaining this concept so effectively.

[@weily-why-lee]

Never change wrath of math, lots of math channels assume too many premises but you stand out cause you mention all the necessary knowledge and steps to get to the conclusion. Never change!

[@WrathofMath]

Thanks so much, Chisala! I'm going to keep doing my best!

[@navster100]

why does this extremely useful video only have 300 views

[@WrathofMath]

Thanks for watching and I am glad you found it extremely useful! I have the same question haha, most of my prob videos do better than that! But there are a lot of videos on Bayes's theorem, so it will take some time for this one to move up in the search rankings. You can help it out by sharing it if you know anyone who would find it useful!

[@ehteshamhassan3861]

Now its 29k😎

[@navster100]

@@ehteshamhassan3861 sheesh

[@topcubasov8942]

@@ehteshamhassan3861 No 50k😎

[@ludwig7223]

Thank you for the incredibly clear explanations! You accomplish the same as a professor in a third of the time, and with much less confusion

[@Matthew-cl8qi]

Very clear explanation of Bayes Theorem. Thank you from a future data scientist.

[@stephaniemorris1706]

I just jumped over here from the law of totally probability video. Taking Stats 315 this semester and you are such a life saver. Thank you so much!!!

[@WrathofMath]

My pleasure, thanks so much for watching and let me know if you ever have any video requests!

[@harkatiyoussef9994]

by far the best video on Baye's theorem

[@WrathofMath]

Thank you!

[@sparrowp2251]

you are a life saviour and so precise witht he concept.Love It Thank You!!

[@valentine7512]

straight to the point, Clear & concise

[@WrathofMath]

Thanks for watching!

[@lanomusambazi8654]

Thank you so much sir for this useful and helpful video. I have understood the language used in Bayes theorem.

[@marianodangelo5908]

Really well explained! Helped me a lot. Could you please make a video on discrete distributions?

[@oliviau.1734]

This has helped me so much you have no idea. Thank you from the bottom of my heart. ❤

[@WrathofMath]

Glad to help, thanks for watching!

[@ajayibasitadeola3352]

The best explanation so far! THANK YOU!!

[@WrathofMath]

My pleasure! Thanks for watching!

[@jim2376]

What you're looking for goes in the numerator and the denominator. The only tricky bit is what is added to the denominator. That would be what you aren't looking for but has the same condition as what you are looking for. Example: looking for the probability of a true positive medical test. (true positive test)/( (true positive test) + (false positive test) )

[@faezehnejati833]

Hey @Wrath of Math :) I deeply enjoyed watching your video(s) on Bayes's Theorem and some other probability videos that you have created. They are beyond awesome! very intuitively explained. Could you maybe produce such good ones on 'Statistics' topics? I miss a full playlist on 'Probability' & 'Statistics' topics in your channel as well. Keep it Up!

[@freduaagyeman2805]

Great video 👍

[@maheebasyed5983]

Amazing explanation ❤

[@WrathofMath]

Thank you!

[@user-en2ct8ql4g]

This is amazing

[@GnarlyGooch]

Cheers mate, good clear video 👍

[@WrathofMath]

Thank you!

[@nopecharon]

Can you please explain how you came up with the denominator

[@adwoamayyy388]

Oh I’m getting it🎉😢❤. I’d need to solve more examples

[@cabbage5114]

This man is doing god's work

[@shaistariaz9783]

Waiting always your videos bro💓💓

[@WrathofMath]

Thanks so much for the support! Let me know if you ever have any requests!

[@harmandeeppal]

I recently started watching your videos, they are well explained. Could you please make videos on Portfolio Standard Deviation, Portfolio Variance, Portfolio Covariance and Portfolio Correlation? Thank you.

[@WrathofMath]

Thanks a lot! I'd love to make videos on those topics, but I can't say it'll be anytime soon. I'll be very busy this fall and winter, so I'm going to mostly stick to topics I've already begun teaching on this channel. But it will be fun to get to some more financial math, and corporate finance eventually!

[@harmandeeppal]

@@WrathofMath That's all right, I can wait. I appreciate the effort you put into making those helpful videos. Thank you for your reply.

[@albertogubernati6677]

Can you make a lesson on weibull distribution with maximum likelihood model? Anyway my sincere compliments for your great and useful videos!

[@lanomusambazi8654]

If you have any videos on random variables please help me. I want to understand it too.

[@georginamoi-dg2sp]

What if the probability of A and B are not given but we're only given the product probability of both A and B? For instance, if the probability of playing soccer and rugby is 0.32 what is the probability of playing soccer?

[@angelaihechu5533]

Thank you so much

[@WrathofMath]

Glad to help!

[@themastmasmashaya7870]

CAN YOU DO THIS FOR ME (A patient visits a doctor suspecting that he may have lung cancer after suffering short of breath. The doctor knows besides lung cancer the patient might be suffering from tuberculosis. He also notes that smoking and exposure to pollution are key causes of both diseases but taking an x-ray would indicate that the patient has cancer or tuberculosis. Given that the patient P(patient is a smoker)=0.3 and P(patient was exposed to low pollution)= 0.9,. Using the Bayesian networks following all steps determine the accuracy of the output from the x-ray results )

[@bilinmezevren-291hzu]

Video was great but there is an error with subtitles, maybe you wanna check again, I informed you... Have a nice day :))

[@ionguzun3952]

Can you please do more videos about Bayes’s Theorem, more examples

[@WrathofMath]

I'll see what I can do, thanks for watching and the request!

[@jimtwisted1984]

Shouldnt the probability of having the disease and not having the disease equal 1? Your sum is .97 so where did the the .-03 go?

[@patrikomelka3918]

ngnl but 32% is still a pretty high chance :D , great viedo really appreciate your help with going through college

[@rahulshah6119]

8:02 isn't the negation symbol ¬? Another great video though!

[@WrathofMath]

Haha, it sure is! Thanks for pointing that out, clearly I was more out of practice than I thought when it comes to probability/logic notation!

[@rahulshah6119]

@@WrathofMath Well if you want more practice, a video or two on markov chains and/or stable matchings/stable marriage wouldn't hurt ;)

[@rightmangolck]

Just wow

[@mevvislenin2784]

Best!

[@WrathofMath]

Thank you!

[@mike_the_tutor1166]

It's spelled Bayes's theorem, according to the AP and Chicago style guides. EDIT: I was right about Chicago but wrong about AP.

[@WrathofMath]

Thank you very much for pointing out that error. I’m gonna hide my face in shame for the next few weeks. I can only hope to be forgiven for still having the error at the beginning of the video! I left the error in the description just once for the sake of other people who make the same error in searching it, though KZbin probably knows exactly what such folks are looking for by now! It has been fixed in the thumbnail and title, thanks again!

[@mike_the_tutor1166]

@@WrathofMath Haha, KZbin probably knows what I'm going to type before I type it. Conspiracy theories aside, if you value visibility over correctness, then I suggest spelling it Bayes' theorem. In my experience, that's the most common spelling, so I would speculate that's what most people would search for. Just speculation though, it may be that "bayes theorem" is the most common search.

[@WrathofMath]

Their search engine and website is pretty darn good at learning! You really notice when it’s completely wrong, but it usually isn’t haha! I like Bayes’s, because as far as I am aware putting an apostrophe by itself at the end of a word to make it possessive is only done for plurals. I’m no English expert, but I’m pretty sure them’s the rules! Admittedly, “Bayes’s” looks pretty ugly, and I am surprised to see that seemingly nobody else spells it that way. Perhaps I’ll change it in a few months and see if it has any effect.

[@mike_the_tutor1166]

@@WrathofMath Oops! I was wrong about the AP style guide! I just looked it up, and it says to add just an apostrophe for proper nouns ending in s. So if you follow AP then it's Bayes' and if you follow Chicago it's Bayes's. For Chicago, you are correct that singular proper nouns are given 's while plural are given just '. I've been confused about this for a while! Now I must also hide in shame!

[@CrannBethadh]

@@mike_the_tutor1166 It is AP that should hide in shame. You have to have something distinct from something belonging to multiple people named Baye. If Johnny and Suzie Baye had created the theorem, then it would be Bayes' theorem, but it was just one dude named Bayes.

[@MrWoaaaaah]

I'm still struggling... please can somehow help me with this example: 1. If someone is white and male, I can detect lies 80% of the time. 2. The chances of someone being white where I live is 81.7%. 3. The chances of someone being male where I live is 49%. What is the probability of me detecting a lie in these conditions? Please can you include it into the formula to help me apply it?

[@nyx211]

There's not enough information in that example to determine P(L). You'd also need to know P(L | not (W, M)).

[@pesofficiall102]

the probability of me understanding is 0

[@asharb5089]

Graph vedios

[@WrathofMath]

Thanks for watching! Do you want graph theory videos? I have a playlist of over 150 of them!