For more information, please see full course syllabus of Statistics

For more information, please see full course syllabus of Statistics

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### Independent Events

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Roadmap 0:05
- Roadmap
- Independent Events & Conditional Probability 0:26
- Non-independent Events
- Independent Events
- Non-independent and Independent Events 3:08
- Non-independent and Independent Events
- Defining Independent Events 5:52
- Defining Independent Events
- Multiplication Rule 7:29
- Previously…
- But with Independent Evens
- Example 1: Which of These Pairs of Events are Independent? 11:12
- Example 2: Health Insurance and Probability 15:12
- Example 3: Independent Events 17:42
- Example 4: Independent Events 20:03

### General Statistics Online Course

### Transcription: Independent Events

*Hi and welcome to www.educator.com.*0000

*We are going to be talking about independent events today.*0001

*We just covered conditional probability and independent events have a lot to do with conditional probability.*0004

*We are going to look at how they relate to each other.*0013

*We are going to actually define what is an independent event is mathematically.*0015

*Then we are going to modify the multiplication rule for a conditional probability for independent events.*0019

*First thing is first.*0028

*Independent events and how that fits together with conditional probability.*0031

*You can think about non independent events, non independent events means that if knowledge of one of the events of the out coming events affects*0037

*the probability that the other events occurs.*0047

*If I know that they are male does it affect my estimate of whether they own a lot or not.*0051

*If I know that this person is obese does it affect my estimate if they have heart disease?*0058

*Those are what we come down independent events, knowing one thing it changes your estimate of the second event.*0065

*Here what you could think about is this, conditional probabilities for knowing that given this person is obese that will change your estimate of heart disease versus*0073

*if you know that this person is not obese these 2 estimates of heart disease will be different.*0092

*They will be not equal to each other.*0104

*That is what we mean by the conditional probabilities are different because if you know one if the conditions that will change your estimate for the other event.*0111

*What are independent events?*0122

*This means that knowledge of one event does not change or affect the probability of the other event occurs.*0124

*Here the conditions of probabilities are the same.*0133

*The probability of heart disease given the kind of car you drive equals red.*0137

*If you drive a red car versus the probability of heart disease, this probability should be the same because car color has nothing to do with heart disease.*0149

*You might say this is independent, car color does not have any varying in my estimate of probability of you having a heart disease.*0172

*Let us talk about this a little bit more mathematically.*0191

*Let us use this example of obesity and gender.*0196

*Here is obesity on this side and gender on this side.*0201

*Does the probability of being male or female change whether you might be obese?*0206

*Does knowing whether somebody is male or female, does knowing their gender affect the probability that they might be obese?*0221

*It turns out that the probability of being obese given that is 20 out of 100, that is the condition of probability.*0229

*We are only looking at the square root.*0241

*The probability of obese given in a female is also 20 out of 100.*0247

*Here you could see that these probabilities equal each other.*0257

*The probability of being obese for male or female are the same.*0263

*How about education?*0272

*Is the probability of being obese given that they have a post high school education is that going to be different than the probability of being obese given*0275

*that they only have a high school education.*0289

*The probability of obesity given post high school is 20 over 100.*0293

*This is my universe is 20/100.*0304

*That probability of being obese given high school only is 30 out of 100.*0307

*It is higher for people who have not have post high school education than people who have had post high school education.*0319

*These conditional probabilities do not mean that this causes this, it is just knowing one fact about these person helps you estimate their obesity probability differently.*0329

*Here you can see these are not equal to each other.*0346

*Let us define what are independent event is.*0354

*We have already talked about this.*0358

*One way to define it is that the probability of A given B is equal to the probability of A given not B.*0361

*Those 2 equal each other.*0372

*It does not matter whether B occurs or not.*0374

*That is what I have written down here.*0377

*There is another way that you could think about this.*0381

*The probability of A given B is equal to just the probability of A in any circumstances.*0384

*This is another way of defining independent events.*0392

*Let us look at that with this data set.*0399

*What is the probability of obesity given male and is that the same with the probability of general obesity?*0401

*Let us calculate that.*0414

*The probability of obesity given male is 20 out of 100 but the probability of obesity over r is here.*0415

*It is obesity of all the people in this sample.*0428

*It is 40 out of 200.*0432

*That is exactly the same proportion 20%.*0435

*Here we see this.*0439

*Being male or obese is independent events in this example.*0443

*Now that we know how to define independent events mathematically let us talk about the multiplication rule for conditional probability.*0451

*Remember those trees that we found was that if you wanted the probability of A and B that is equal to the proportion of A given B multiplied by the proportion of the probability of B.*0462

*Think about these spaces.*0484

*B and what proportion of that is A given B if you multiply this together you will get that raw score.*0486

*This is what we call the multiplication rule for conditional probability.*0494

*Out of this you could also get the definition of conditional probability where probability of A given B equals the probability of A and B over the probability of B.*0498

*We already know the multiplication rule and that is just one step around.*0516

*I should have to say here that obviously you could have probability of B given A × p(A) because you always want to have that entire world that you are living in.*0520

*The condition that you are living in.*0536

*Independent events we now have a slight change than this because the probability of A given B equals the probability of A, look at this rule again.*0538

*All we have to do now is this, in order to find the probability of A and B since this equal this we can now just do probability of A × p(B).*0550

*This is exactly equal to the p(A) given B.*0570

*For independent events we can simplify this.*0575

*This all goes back to the multiplication rule.*0578

*In independent events, now you could just write p(A) and still be able to calculate p(A) and B.*0582

*The other way that you could think about this is you could change it into figuring out different relationships among these things but you can also generalize it to more than just 2 events.*0593

*We could put 3 events, p(ABC) = p(A × B × C).*0611

*You could do 4 events, 5 independent events because you can do this infinite times.*0624

*That way I like to think about this is going back to the sample spaces thinking about this independent events as slots that you could fill.*0633

*Let us think about flipping a coin, those are independent events.*0643

*Knowing that you first flip is a head does not do anything for my next flip of coin.*0646

*It is still a 50-50 chance of getting heads.*0654

*Here you could think about this as the probability of A, probability of getting heads is 50%, 50%, and 50%.*0657

*You could see that it will go on and on and on.*0665

*Flipping coins are classic examples of independent events.*0668

*Let us move on to some examples.*0673

*Here is example 1, suppose you draw a card at random from a deck of cards which of these pairs of events are independent?*0675

*You are just drawing 1 card and just because they say events it does not mean you are drawing 2 cards.*0682

*It just means that it is 2 different aspects of cards like heart and jack.*0689

*Here it says it is getting a heart independent of getting a jack.*0695

*Does having any of the one affect the probability of getting the other?*0701

*We could line out the rule for independence of events.*0704

*Probability of heart given jack should equal the probability of hearts.*0712

*Is that true? Let us think about this.*0720

*There are only 4 jacks, that is my whole universe and the probability of getting a heart is ¼.*0724

*That is the probability of getting a heart overall.*0730

*I would say these are independent.*0734

*I chose probability of heart given jack but you could have also done it the other way around.*0736

*Probability of jack given heart is that equal to the probability that you will just draw a jack.*0744

*The heart world is 13 cards so out of 13 there is only 1 jack that is 1/13.*0752

*The probability of drawing a jack is 4 out of 52 which is 1 out of 13.*0762

*Eventually we will get out of 13 and we will see that it does not matter which event you pick as your condition are independent.*0770

*Are these 2 events independent?*0781

*Getting a heart or getting a red card.*0785

*We could set that up again heart versus red card, heart given red card, is that the same as the probability of getting hearts overall?*0788

*We already know this one, it is ¼ same as here.*0798

*There is a probability of getting a heart given that you already have a red card is going to be different.*0802

*Half of the cards in the deck are red, hearts and diamonds.*0807

*That is 26 cards.*0813

*Out of these 26 cards half of those are hearts, 13 out of 26 are hearts.*0817

*That is half of those cards are hearts if you know that is a red card.*0828

*½ is not equal ¼.*0834

*I would say these are not independent.*0836

*Here let us say independent and here not independent.*0839

*You could always test it the other way as well.*0847

*Probability of 1 given heart is that equal to probability of getting a red card?*0850

*What about this last one, the probability of getting a 7 given heart is that equal to just getting a 7?*0856

*Let us see.*0869

*The probability of getting a 7 is that there a 4 7’s one for each suit out of the 52 cards.*0871

*4 out of 52 and that is going to reduce to 1 out of 13 because for every suit there is only 1 7.*0880

*What about probability of getting a 7 given that it is a heart?*0893

*If it is a heart that is only 13 cards but the probability of getting a 7 is 1 out of 13.*0899

*These are equal.*0905

*Let us say independent.*0907

*Here is example 2, the US department of health and human services found that 30% of young Americans 18 to 24 years old do not have health insurance.*0913

*If you sampled 2 young Americans at random what is the probability that the first has insurance and the second does not?*0922

*At first you might think this is sampling without replacement.*0930

*You might think that this is conditional but if you are sampling from the entire US because it is just 2 young Americans*0935

*at random it changes the probabilities into tiny decimal amount that it does not matter.*0944

*We could treat this as almost independent event.*0957

*Frequently that is one way that independence is used for almost independent events where it might affect it slightly.*0961

*Think about drawing one young American what is the probability that any 1 young American would not have health insurance?*0978

*That is 30%.*0987

*What is the probability that drawing 1 American has health insurance?*0990

*Here is the first guy, has health insurance.*0995

*That will be 70%.*1001

*You can multiply that by the probability of the second guy not having health insurance.*1003

*That is .3 or 30%.*1013

*If you multiply those together then it says 21% chance that you will get the combination that the first guy has insurance and the second no insurance.*1017

*Remember we noticed because of the revised multiplication rule where we can just look at this as being equal to the probability of the first guy having insurance times*1039

*the probability that the second guy has no insurance.*1055

*Example 3, a state school gets 1725 applications, are being admitted and going to private school independent events?*1064

*We could apply our definition of independence.*1075

*Is the probability of being admitted given private school?*1080

*Is that equal to the probability of just being admitted?*1092

*Let us check.*1096

*Here is the probability of being admitted given private schools, that is this university right here.*1098

*That is going to be 220/483 that is my probability of being admitted given that it is a private school.*1105

*This the probability of being admitted at all.*1116

*This is 870/everybody and we want to know are these equal to each other?*1118

*I’m just going to use Excel as my little calculator.*1134

*220 ÷ 483 that gives us about 46% chance of getting in if you go to private school.*1140

*870 / 17.25 is a slightly higher chance of getting in.*1156

*This is probably not true, you have a chance of getting in if you go to private schools.*1175

*That is 46% is not equal 50%.*1180

*I would say that it is small but there is a slight difference between being admitted and these are not independent because*1185

*there is a slight difference in the conditional probability versus the overall probability.*1196

*Example 4, about 11% of college freshman have to take a remedial course in reading, suppose you take a random sample of 12 college freshman from around the US,*1206

*what is the probability that none of the 12 have to take remedial reading?*1216

*What is the probability that at least 1 has to take a course in remedial reading?*1222

*Here we could use the multiplication rule because we could assume almost independence in picking 12 people, it is almost like sampling with replacement.*1228

*It is not going to affect the probability that much.*1245

*What is the probability that the first guy does not take remedial reading and you want to multiply that by the probability that the second kid does not take remedial reading,*1248

*all the way up to the probability that the 12 kid does not have to take remedial reading.*1274

*It is not 11%, if you draw a percent random there is 11% chance that this college freshman has to take remedial reading.*1282

*The flip side of that not having to take it is 89%.*1293

*That would be 89 × .89 × .89, 12 times .89 ^{12}.*1298

*That would be .89 ^{12}, 24.7%.*1313

*That 25% of students of this sample, if we took a group of 12 people, 25% of the time all 12 do not have to take remedial reading.*1330

*Notice the probability that at least 1 have to take a remedial course.*1345

*We should not apply this rule because we do not know which one of these guys takes the remedial course.*1352

*We do not care which one.*1364

*We do not care if it is the first or second, or the first and third, or the first, second, third, or all of them.*1365

*Except for the last guy that do not have to take.*1377

*We just want to know, what is the probability that at least 1 will have to take remedial course?*1384

*That is every combination.*1392

*1, 2, 3 all the way up to 11.*1393

*The only case that you want to leave out is when all 12 do not have to take a remedial course.*1398

*What we could do is 1 – the probability that all are exempt from remedial reading.*1407

*We already know that it is 1 - .247.*1424

*That should give us .753.*1431

*That should give us about 75% of samples of 12.*1438

*The samples are at least 1 where they have to take a remedial reading course.*1446

*That is our shortcut.*1452

*That is it for independent events, thanks for using www.educator.com.*1462

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