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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

Transcription: Independent Events

Hi and welcome to

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 affects0037

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 versus0073

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 .8912.1298

That would be .8912, 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 1462