Basics of Probability: Unions, Intersections, and Complements

An introductory discussion of unions, intersections, and complements in the context of basic probability. I include a discussion of mutually exclusive events, as well as the addition rule. I work through a simple die rolling example, and also an example of a scenario where we randomly sample from a population of people, where some have diabetes and some have hypertension. The various scenarios are illustrated with Venn and Euler diagrams. (The percentages given in the video are loosely based on statistics for 50 year-old Canadian males.)

Thank you so much. I don't understand how my professor can be so bad at teaching this content, whereas this method of presentation and explanation makes it interesting again.

[@thomaswoodall4505]

How lucky am I to have you as a proffesor. Im currently in your DE STATS course and would seek out youtube explanations on some content only for my own proffesor to be the creator the exact type of content I need. I hope you seriously understand the impact of your work. To offer this level of education for others free of charge that do not have the luxury of being your student is so thoughtful. Wish me luck on the test tomorrow!

[@jbstatistics]

Thanks for the very kind words! I'm glad to be of help! I do make these primarily for the benefit of my own students, so you're the target audience :) I hope the test goes well tomorrow! (For both of us!) Once again, thanks for the very kind words.

[@user-jj4zb5jd6r]

🥺100% it's very rare to find profs that make you feel like they genuinely care about your education

[@sophie5684]

This is the best stat video ever. It's so clear and helpful. Thank you!

[@jbstatistics]

Thanks for the very kind words!

[@ramkotha4726]

OMG!! Is this real.. This is the best thing that could happen to me in my journey to learn data science, in the last 8 months. What an explanation, examples, and visuals.. a BIG thank you to Prof. Jeremy Balka. I'll remember this channel for the rest of my life. I wish there are calculus, linear algebra and other videos, but it'd be too much for asking.

[@nicetomeetchuu3267]

niggä

[@codecutner4902]

my textbook structured this topic horribly... this video helped a lot

[@Sum_random_person]

i still dont get it

[@cartilo2619]

Thank you, I had a hard time understanding the concepts in class but you made it so easy to finally grasp!

[@kobebud1464]

YOU SAVED MY LIFE

[@The3497jsm]

One of the best and simplest explanations.. thank you for the content

[@linesup6417]

This explanation is millions times better than my uni statistics teacher explanation❤️ Every time he used to say you are fail you are fail. He is so... I don't know what to say him but you saved my life bro ❤️

[@eyobhaile9875]

your videos are short and clearly understandable , good job.

[@AyushiSharma-d3u]

best video cleared all the concepts thank you so much

[@harveyspecter3361]

Your videos are amazing, never stop making them!

[@MrMukulpandey]

Beautifully explained. Try to add real world example in every concept because it keeps in mind for long time.

[@rayped2771]

You deserve a raise

[@viac795]

I didn’t understand the concept until this video, thank you for making your explanations so easy to understand and the several examples really helped

[@jbstatistics]

You are very welcome! I'm glad to be of help!

[@apg_skywalker_2014]

Thank you JBstats. Your explanation is so crystal clear.

[@nicetomeetchuu3267]

sup nigga

[@user-wr3nx9wu2c]

Great work professor ❤

[@neviswarren]

Excellent. Thank you.

[@CloroxBleach-ox7cf]

Really great illustrations and explanations, thanks for this video!

[@Shumayal]

So I have another stats course in my masters program and I am happy to refer back to my tutor who taught me stats and probs in my undergrad. Glad to see your new videos! I was wondering if you can make a logically ordered playlist from 0 to the end? Sorry if you have already arranged it, just wanted to confirm.

[@joshbrown8291]

Thank you for the great explanation!

[@jbstatistics]

You are very welcome!

[@alex_f_stop]

Hey Jeremy, thanks a lot for all your stats & proba content!! I've been watching your videos for a while, and I'm glad to see you posted new content. What I like most about your videos is that you're trying to break down the technical stuff. This is really hard to find, a lot of online educators are just trying to give intuition these days - while this is nice and entertaining, I don't find it too helpful ultimately.

[@jbstatistics]

Thanks Alexandru! I'm glad you've found my videos helpful. I do try to give the real deal here -- including both intuitive explanations and the necessary technical details. All the best.

[@user-in2ws8cq5n]

I have learnt something. Thanks alot

[@wuutchillll]

THANKYOU SOOO MUCH! I'M GOING TO REPORT THIS IN MY CLASS 💕

[@keithdube5955]

YOU THE MAN🙌🙌🙌!!!!!

[@thedeathofbirth0763]

You are brilliant, I beg for more examples.

[@MISHUCHOPRA]

best explanation. Thankyou

[@pittbull]

Thank you so much!

[@emigenbritz7238]

Thank you so much!!! 😩

[@kevinschaeffer4786]

Another outstanding video! Thank you sir

[@bluesy666]

Thanks for the great content as usual

[@jbstatistics]

You are very welcome, and thanks for the kind words.

[@manudevi4287]

Awesome helped with the basics 🙂💌

[@driosemelda4480]

Wow... You made it that easy ...thanks 🙏

[@w3w3w3]

thanks this is a great playlist!

[@azaniftikhar8761]

great video

[@azaniftikhar8761]

I agree

[@mayurtasre7378]

Thank you so much you made probablity easy for me☺

[@zachstandley6770]

You've explained probability better than khan academy.

[@jbstatistics]

Ain't no "probably" about it :)

[@Aakash_Goswami1]

Wait a sec... I thought I was watching Khan Academy

[@syedrizvi2687]

Thank you:)

[@paulhetherington3854]

Really awesome, articulations verses, Locative of ignorances, by LAN / WAN = WiFi!

[@andyandrw]

Nice video!

[@mohfa1806]

Thank you for your great videos... I have a question please : when calculating P(E n F) you said it is 1/6 , but then i thought since events E and F are independent since are rolling the dice independently each time , then P(E n F) = P(E) x P(F) = 1/2 X 1/2 = 1/4 !. Am i missing something here ?.. Thank you in advance

[@jbstatistics]

Those events are based on the roll of a single die. (Thus leading to the given sample space of S = {1,2,3,4,5,6}.) The fact that we often assume repeated die rolls are independent is not relevant here. On a single roll, the probability of getting an odd number on the top face (event E) is 3/6. The probability of getting a 4, 5, or 6 on the top face (event F) is 3/6. The probability the number is both odd and a 4, 5, or 6 (event E n F, which is just the number 5) is 1/6. E and F are not independent.

[@mohfa1806]

​@@jbstatisticsThank you for your fast reply , i really appreciate it.. It is semi clear to me now , but i still can not understand how event E occuring will effect the probability of of event F as we are rolling the die independently each one time ? . This all starts with me when i tried to solve P(E n F) using multiplication rule for indepndant events... Iam sure you are of course correct saying E and F are dependent but my problem is that i cant understand how they are dependent since each roll of a die is seperate !. Thank you in advance

[@jbstatistics]

@@mohfa1806 " i cant understand how they are dependent since each roll of a die is seperate !." I don't know what to tell you here, as in my first response I addressed this as well as I can. It does not matter, at all, if the rolls of a die are considered independent. We are talking about a single roll. The die is rolled. We look at the number on the top face. It is one of the numbers 1, 2, 3, 4, 5, or 6. What's P(E)? 3/6. How about P(E|F)? Well, if the number is a 4, 5, or 6, there is a 1/3 chance it's odd, so P(E|F) = 1/3. P(E) does not equal P(E|F), so the events are not independent. If we assume that repeated die rolls are independent, that does not mean that all events with associated with rolling a die are independent. Is getting an even number independent of getting an odd? No, of course not, since if we roll an even number it cannot have been odd. Is getting an even number on the first roll independent of getting an odd number on the 40th roll? Sure, but that's not what we're talking about in this video. I address conditional probability and independence in detail in other videos. This video is about the basics of unions, intersections, and complements.

[@mohfa1806]

​@@jbstatisticsthank you prof. For your responses and being patient with me , i really appreciate it. And sorry for my late response to you since i was little ill in the past two days. There is still something that confuses me : When we say P(E I F) , then according to my little knowledge it means : "the probability of event E such that event F occured" . My question is : when event "F occured" , then one number occured , either 4 or 5 or 6 , as they can not occur all together in one roll at the same time. This leads me to confusion , as according to my logic P(E I F) can be equal to either 0/6 if event F was 4 or 6 , OR equal to 1/6 if event F was 5 , and since we can have two answers then this proves that events E and F are dependent . Is my logic correct ? . Thank you again for your patience

[@yasmeenalajmi1091]

Thank you so much I actually understand stat than just memorize

[@jbstatistics]

Great! I do try to teach the real deal, and very much hope that students try to learn and understand rather than memorize. All the best.

[@kelpsalad1330]

I get it now!!! Thank you!!!!

[@Cleisthenes2]

You count the intersection twice, so you have to subtract it once

[@Victual88]

your a hero mate!

[@englishwithravi4590]

Nice video...thank you👍💕

[@rushi5650]

Like That. I understand all

[@jaclyn2001]

This helped me so much!!:)

[@cheezey3295]

Hey man I'm begging you to make a video on combinatorics that's one of my worst areas in stats lol. My finals are in a month so I don't have much time...

[@lundusimuchenje762]

life saver thank u very much

[@jbstatistics]

You are very welcome!

[@gazee1495]

يعطيك العافيه

[@doobas2171]

dont the total probabilities have to add up to 1? Or is that only for the sample points and not the events?

[@jbstatistics]

Only the probabilities of the sample points need to add to 1. Probabilities of events can add to any nonnegative value (under the restriction that each individual probability can't be greater than 1). It is very common for two events to have probabilities that add to something greater than 1.

[@doobas2171]

@@jbstatistics thanks!

[@phlufasaal-hasan6530]

This video was extremely helpful! However, I have a question. At 13:48, why is it a union of the compliments and not the intersection? The reason I say this is because, the probability of not having diabetes would be the green region plus the probability of someone only having hypertension. While the probability of not having hypertension includes the green region plus the probability of only having diabetes. So wouldn't the probability of not having diabetes and hypertension, be the intersection of not having both? Sorry for the long winded question, and thanks in advance!

[@abercrombie4ever100]

thank you for this video! i actually understood it! thanks to you:)

[@perceptrongaming4290]

excellent!

[@cococnk388]

Thanks a lot, Z = "Remaining un known event of the sample space" {D,H,Z} are events of the sample space S At first, this is what I did P(S) = P(H) + P(D) + P(Z) =1, P(Z) = 0.6 (you got a 0.67) but i later on realised that this only applies if the events (Z,H,D) are mutually exclusive. P(S)= P(H u D u Z) P(S)= P(H)+P(D)+ P(Z)- P(H n D) - P(H n Z) -P(D n Z) - P(H n D n Z) we donot have all the probability values of the intersection terms, so you decided to use the "complement" in order to find P(Z) : P(D n H' ) = P(D) - P(D n H) = 0.03 P(H n D') = P(H) - P(H n D) = 0.23 P(Z) = P(S) - P(D n H) - P(D n H' ) - P(H n D') P(Z) = 0.67 and not 0.6 as I thought. Illustrating it on a vein diagram is priceless, great work sir. If someone has something to add or a different approach of what I just did above, kindly let me know in the section comment of my comment.

[@Justajawnie]

@jbstatistics if you are still responding to your videos, would love to hear your thoughts on a good textbook. I would like to get a good basis in probability and statistics for the purposes of creating data models and eventually understanding AI.

[@vijithasenadeera7576]

thanks very good .good luck

[@jbstatistics]

You are very welcome!

[@krushnakantdeshmukh2188]

Perfectly explained

[@suzuori3430]

you explain it very well. thanks

[@maroofali2142]

Hey man...i have found ur videos really helpful...thanks for all the good stuff...n i have a doubt...it would b great if u cud just reply me here... How do we decide that whether our problem is to be solved using the binomial or Poisson distribution.??...thanku

[@jbstatistics]

That's a big question that's not possible to answer in a reasonable fashion in a comment. I have extensive video support for both of those distributions, and an overview video on discrete probability distributions that illustrates similarities and differences between a variety of discrete probability distributions. Those videos might be helpful to you.

[@maroofali2142]

jbstatistics thanks my man...really appreciate the fact that u took time to leave a reply...okkk...i'll check those videos out...thanku again...i cant tell u how much time these videos of urs have saved for me...have a good day/night Wherever u r

[@liftyshifty]

can you please explain why p(a&b) = p(a|b)/p(b) and also equal to p(b|a)/p(a)

[@jbstatistics]

It's probably best to view my conditional probability videos, and see if that makes sense to you after. I walk through a visualization of the conditional probability formula in my "Introduction to Conditional Probability" video. If the conditional probability formula makes sense you to, then a reworking of it (as you have given) should make sense as well.

[@henkrishna8010]

Concepts are Chrystal clear

[@jdamayadaw-as6589]

How to get the 0.03 in the part oh D' and H

[@young5395]

If only I would have found you a week earlier.

[@uzairrashid7724]

it was really helpful, thanks 😁

[@jbstatistics]

You are very welcome!

[@andilemasela4668]

what textbook can i use for this chapter

[@saadb0ys]

I freaking love you dude

[@user-es7xx8bg8m]

at 8:08 why is the probability of F union G = 1/6 and not 1/5? I understand the 6 must be the possible outcomes of the dice, but aren’t we looking at just the sets F and G, and not S?

[@jbstatistics]

The sample space is the set of all possible outcomes. Precisely one sample point from the sample space will occur. There are six sample points: 1, 2, 3, 4, 5, 6. The intersection of F and G is one of these sample points, the number 6. F n G = {6}. Since the sample points are equally likely in this scenario, P(F n G) = 1/6. I'm not sure why you think we should be tossing away a sample point.

[@Victual88]

p.s do you have a patreon? (:

[@juniorakimr]

How did you get 0.67

[@rababkarim7469]

Thank you 🙏🏻

[@joshman1019]

You should be the person writing for ZyBooks because they leave out 90% of the important explanations and just hope that you can assume what is happening.

[@harryfriedman361]

I love u

[@BanaNa-db5xf]

I'm trying to focus on the content, but your voice is so distractingly pleasant.

[@jbstatistics]

I'm not sure what to make of that overall :) But I'll take it as a nice compliment!

[@BanaNa-db5xf]

@@jbstatistics Your voice would be nice for ASMR videos. :)

[@leprechaunbee3381]

It does the opposite to me. I actually focus more. His voice is so nice. Sal Khan sounds annoying af.

[@marcpaulglimada3332]

Why is 0.90 the answer in D'? Where did you get that answer?

[@deliberativedisciple]

Because...if you have a 10% chance of having diabetes, it also means you have a 90% chance of not having it.

[@sandybattulga6619]

LEMME GIVE YOU A KISS YOU BEAUTIFUL PERSON. SO GOOD AT EXPLAINING STATS.

[@jbstatistics]

Thanks! I'm glad to be of help!

[@uttamraut7577]

8:07 probability of p interction g is 1/5

[@hectortorres1222]

I noticed that too.

[@abdullahalseflan2128]

Why are the three events mutually exclusive, though?

[@jbstatistics]

What 3 events?

[@duckymomo7935]

are we going to do sigma-algebras?

[@jbstatistics]

No, I won't be including a discussion of sigma-algebras. I'm covering probability only at the level of an applied introductory statistics course for non-math and non-stats majors. I might go a touch beyond that at times, but only as the topics relate to applied statistics topics.

[@duckymomo7935]

jbstatistics but your image is exactly a sigma (o standard deviation and talking about sets with unions and intersections)

[@jbstatistics]

I missed that connection at first :) The sigma in my logo represents variability, and not anything from formal probability theory. I'm an applied guy -- just trying to make good decisions under uncertainty.

[@Sum_random_person]

i still dont get it ಠ︵ಠ

[@namonscreen]

Why probability of P(E)*P(F) is not equal to P(E and F)???

[@Mpendulo-ox3sb]

These are not independent events

[@fishertech]

why is P(E∩F) same as P(F∩G)? EF has 6 sample points and one common point which gives 1/6, but FG only has 5 sample points and one common point so why is that equal 1/6

[@fishertech]

7:45

[@jbstatistics]

There are 6 sample points in the entire sample space. E and F share a single sample point. F and G share a single sample point. Since the sample points are equally likely, this implies that P(E n F) = 1/6 and P(F n G) = 1/6. (If one of those 6 equally likely sample points is randomly picked, there is a 1/6 chance it is a 5, and a 1/6 chance it is a 6.) E U F = {1,3,4,5,6} and F U G = {2,4,5,6}, but that's irrelevant as far as P(E n F) and P(F n G) are concerned.

[@user-hp1bi7se4x]

I understand this stuff, it took me a few look over of notes and this video, but seriously whoever invented this part of stats was very bad at describing things. The wording everyone uses is so confusing and is made to seem harder than it is. This needs to be reformatted bad

[@jbstatistics]

I'll disagree with you strongly here. There may be some spots in stats where historical language choices are in retrospect not the greatest, but here? Unions, intersections, complements? The usage of all these terms is very consistent with their use in common English. The symbol for union even looks like a U. The symbol for 'the other one' is just that flipped. If you think this terminology is terrible, you must have an alternative in mind. What do you suggest?

[@user-hp1bi7se4x]

@@jbstatistics no, not that part of it. When people start explaining it saying unions can be A or B or Both and intersections are A and B (isn't that both as well?). Just as an example, I know we can't have 2 types of blood, but for a not mutually exclusive event why would I use intersection instead of union? Simply because I know the two types of blood thing? So doesn't that add common sense or even knowledge into the equation (atleast when interpreting someone else's data)? Thats where the wording was confusing to me as. I came here and you didn't keep saying the both with union so I took it out of my thoughts and kept it with just "or". Maybe I'm just missing something lol. Thanks for the fast response too, isn't this video 7 years old?

[@jbstatistics]

@@user-hp1bi7se4x In the English language, "or" can be the inclusive or (either or both) or the exclusive or (either but not both). I state very explicitly in this video that when we're using the term "or" in probability we mean or in the inclusive sense. "When we use the term A or B in probability, we are referring to their union, and using the word "or" in the inclusive sense; A or B means A or B or both." I realize this can be a point of confusion, so I get miles ahead of it and address it explicitly and as clearly as I am capable of or ever will be capable of. If I keep going on about it every time I encounter the word "or", it would be a mess. Sure, there are times when we might need to use "or" in the exclusive sense, and then we're clear about it "A or B but not both." I really don't understand what your point is about the blood types, but no, there is no ambiguity or potential discussion about whether "mutually exclusive" should involve the union vs the intersection. Two events are mutually exclusive if they share no sample points. If they share no sample points, the intersection is the empty set (there's nothing in there), leading to a probability of that intersection of 0. If two events are mutually exclusive, the knowledge that one occurs tells you that the other one did not occur. "Mutually exclusive" has pretty much the exact same meaning in ordinary English and in probability. These things cannot happen together; they are mutually exclusive. The video's 7 years old, but I'm often logged in and get notifications. I address some of them, and don't check to see when I made that particular video. Sure, it was 7 years ago that I made this, but we'll be using this same terminology in 50 years, and these concepts will still be the same.

[@user-hp1bi7se4x]

@@jbstatistics I understand it, your video helped me have that eureka moment, but I still find it overly confusing for people. I'm not sure how it should or would be reworded but it's weird when we can have an example like what's the probability of both event A and B, having "both" and "and" but the big thing is the and, it's just in many descriptions is also says both for unions (as it does measure it as well) so that could cause almost unneeded confusion with a question worded like that. Also I just saw you're a professor, please don't eat me alive here lol. I'm having this conversation in my whatsapp group for school too, and saw many others having similar confusions, I had to go over the material 3 times almost and watch your video 2wice before I was like alright I'm overthinking it (which could be the confusion issue to be honest) Lol sorry I know my blood thing is confusing (ironic lol). Well thank you for the crazy fast replies and I really appreciate your videos

[@ilsignorgiappone640]

Molto interessante. Suggerisco anche versione in italiano da "corsi consulenze NPR" kzbin.info/www/bejne/eqfXe6uenb2dbNE

[@henkrishna8010]

Concepts are Chrystal clear