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Lecture Comments (1)

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Post by Saadman Elman on June 10, 2015

Thanks it was very helpful.

Addition Rule for Disjoint 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 0:08
    • Roadmap
  • Disjoint Events 0:41
    • Disjoint Events
  • Meaning of 'or' 2:39
    • In Regular Life
    • In Math/Statistics/Computer Science
  • Addition Rule for Disjoin Events 3:55
    • If A and B are Disjoint: P (A and B)
    • If A and B are Disjoint: P (A or B)
  • General Addition Rule 5:41
    • General Addition Rule
  • Generalized Addition Rule 8:31
    • If A and B are not Disjoint: P (A or B)
  • Example 1: Which of These are Mutually Exclusive? 10:50
  • Example 2: What is the Probability that You will Have a Combination of One Heads and Two Tails? 12:57
  • Example 3: Engagement Party 15:17
  • Example 4: Home Owner's Insurance 18:30

Transcription: Addition Rule for Disjoint Events

Hi and welcome to www.educator.com.0000

We are going to be talking about the addition rule for disjoints events.0001

First, we wanted to go over again what disjoint events mean?0005

and then we will talk briefly about the meaning of the word or because it has a slightly different meaning in statistics than in regular life.0015

Then we will talk about how to calculate the probability of a or b and that method is going to be the addition rule.0023

We are going to talk about the addition rule when A and B are disjoint events, as well as when they are not disjoint events.0032

Let us talk about disjointed events.0041

remember we have said before disjoint events are mutually exclusive and they cannot both happen at the same time.0046

so you cannot draw one card from a deck and have it be both a Jack and Ace.0053

It has to either be a Jack or Ace, you cannot have it be both at the same time.0058

Where can you select a student at random from high school and get both the junior and senior.0065

We have to be just one or the other.0072

However, you can select a card from a deck and have that it be either a jack or a heart.0074

they are not mutually exclusive.0082

They can both happen at the same time.0085

They do not always have to but they can.0086

if we wanted to draw this idea as a picture we may show these two events this is the sample space of all the possible things in the world that might happen.0089

Here is when A is true, your card is a Jack.0100

Here is when B is true, your card is an Ace.0105

Notice that there is no overlap between those two things.0109

There is no part of this space both the Jack and Ace parts are true.0113

In this picture we show a non disjoint events.0118

Here we might have as the space for all the events when jack is true but here we have the space where the card is a heart.0123

Here we have this space where it is both a jack and a heart.0134

Here we see that it is possible to be both jack and heart.0142

Here we might have jacks and hearts like jack of cloves or spades.0146

Here we have hearts that are not jacks, the king of hearts or the ace of hearts.0155

Now let us talk about the meaning of or.0157

In regular life, we usually use the word or like this, would you like soup or salad?0163

Would you like it to be red or blue?0170

Typically we mean would you like one of these things or the other one?0173

You cannot have both at the same time.0179

You can only choose one.0181

If you say you like this you mean to say you have to pay for the other one.0183

How do in math, statistics, and computer science, we are not talking about choices.0190

We are talking about truth values.0196

A or B means that either A is true or B is true, or they are both true A and B.0199

Basically the big idea of this is that at least one of these is true.0211

If both are true then you fit the world, at least one of them is true.0219

In that way or means slightly different in statistics lingo.0227

Now we could talk about how to calculate the probability of A or B and that is called the addition rule.0233

First let us talk about the addition rule for disjoint events.0244

The way you could think about this is what is the probability that you will land in this space or this space?0247

What is it asking you what is the probability that you will land in one of this shaded areas out of all the other possibilities?0259

In that way you might see that you might add together these 2 probabilities.0273

If A and B are disjoint, we need to first talk about what is the probability of A and B is true?0278

You could see that there are no points here.0290

There are no space, they are both A and B are covered in.0290

That overall space between A and B and because of that we would say 0.0298

There are 0 part of this space where both A and B is true.0306

We only have A being true or B being true but not both.0310

Here we have to calculate what is the probability of A or B.0315

You could easily see, you might want to add together probability of A and B.0321

This is 25% and this is 25%, maybe we have 50% likely having A or B.0329

That only works for disjoint events.0338

Here if it is disjoint events the world is simple.0349

If we only have one disjoint events, let us imply that same logic.0351

Remember we have talked about disjoint events we are asking what is the probability that you are lined within these shaded event spaces.0357

P(a or b) means what is the probability in one of these shaded spaces.0367

It is the same idea in non disjoint spaces.0373

I want you to watch carefully.0378

It is what the probability landing in A or in B.0380

Notice that this space is counted twice.0386

When you say p(a or b), what you want to know is this 8 shaped area and we do not want to count that part twice.0391

How can we come up with a general addition rule that will work for either non disjoint events.0408

You might want to start off thinking about this in the same that we did before.0414

First it might be helpful to add together the probability of A and B but we have counted this part twice.0422

It will be helpful to take one of those out.0435

We might subtract the probability of A and B.0438

In that way we can use one of these out and I will get just this area.0444

The reason why this is called the general addition rule is because it actually works for disjoints events as well.0450

Let us try for disjoint events.0460

We have probability of A or B, and we have probability of A + B, we will subtract the probability of A and B.0463

Here what is the probability of A and B?0482

It is 0 here.0485

What we are doing is adding these probabilities and subtracting 0.0487

That means the addition rule for disjoint events.0499

This general addition rule works for all events, both disjoint and non disjoint.0503

The general addition rule is important and very useful method of contact.0511

There is a different way you could write it.0520

I just want to briefly show that to you here.0524

I’m just writing down the generalize addition rule and rewrite this using slightly different notation.0527

Many of the frequent notation you might see is the use of union and intersection.0538

Union this goes to or.0544

When 2 countries unite, you are thinking about 2 pieces coming together and now both pieces count.0554

That is the picture that you should think about or.0568

When Alaska is united to the rest of United States together.0571

It is or whether here or here.0581

The other piece of notation you need to know is the intersection notation.0587

This one match perfectly to and.0597

The idea is where do they intersect?0602

Where A intersect with B?0608

We could just rewrite this using this notation.0610

This is often used for sets.0614

You could just use this notation.0616

Instead of or we can use union A and union B and that would mean the probability of A and B.0619

We do not have to change anything there - the probability of A intersection B.0630

This is just saying this whole area shaded in = p(a + b) and take out this little section.0637

Let us move on to some examples.0648

Let us say that I have chosen a person at random, which of these are mutually exclusive?0653

By using mutually exclusive you want to think of the same term as disjoint events.0659

They will be used interchangeably.0669

Has ridden a roller coaster or Ferris wheel, does one prevent the other from happening?0673

There are people who have both ridden roller coasters and Ferris wheel so I would say that these are non disjoint or not mutually exclusive.0681

There are nothing that prevents people from doing both.0692

I think I can have both songs on iTunes.0701

Has brown eyes has brown hair?0708

Here too you could have both brown eyes and brown hair or you could have just brown eyes but other colored hair.0712

You could just have brown hair and other colored eyes.0720

This is also non disjoint.0723

Is it left handed or right handed?0728

Your dominant hand is just one or the other.0732

I would say this is probably disjoint.0739

What pops into my mind right now are the ambidextrous people but even them they prefer one hand versus the other.0745

That is just disjoint for now.0756

Has had chicken pox never had chicken pox?0760

This one is definitely disjoint because we cannot have both have it and not at the same time.0765

Suppose we flip a coin 3 times what is the probability that you will have a combination of 1 head and 2 tails.0776

You have to remember the lessons from last time as well as the addition rule.0786

First we have to figure out what the entire sample space looks like.0792

The whole square would be, for example if we toss the coin 3 times that would mean that is 2 different power of 8.0797

First flip and second flip, third flip.0811

Half of these 8 are supposed to be heads and half of them are tails.0816

I am just going to start off with that.0822

Second one, half of these have to be heads and half have to be tails.0824

Half of these have to be heads and half of these have to be tails.0834

Same thing here.0847

I do not see the pattern but it is just cuts in half every single time.0848

The last one half of 2 alternates.0856

Now we know that each of these are probability of 1/8.0862

We can use the addition rule to figure out what is the probability that you have a combination of 1 heads and 2 tails.0870

We could just find other one heads and 2 tails and we add those probabilities together because these 3 things are disjoint events.0878

You cannot have both heads-tails-tails, and tails-head-tails at the same time nor can you have tails-tails-head at the same time either.0894

We have the probability of 1 heads and 2 tails = 3/8.0904

Ann and Bill are planning their engagement party, the room will hold 200 people so they agreed that Ann will invite 100 friends and Bill will invite 100 friends.0916

Everyone invited in the party showed up but only 140 people turned up, what must have happened?0928

It might be helpful to think about this as a picture.0939

Here all the people that Ann invited and here all the people that Bill invited, and together it is going to be 140 people.0943

There are some portions of difference that must have had overlapped.0954

We are trying to figure that out.0962

Right now we are looking at frequency instead of percentages, but obviously you can turn these to percentages if you want.0964

I’m just going to keep it in frequency form.0974

We are looking for the probability of A or B – the probability of A and B.0976

You can turn this probability statement into the total number of people.0994

We are just going to keep it as this.1013

We have 100 of Ann’s friends, 100 of Bill’s friends, and we do not know the overlap.1017

We do know that 140 people showed up eventually.1026

We do know that part.1034

The way you could do this is to divide everything by 200 because that is the total number of invited.1036

If we do this and we have 200 – p( a and b) I want to add this to both sides so that I can make it positive.1048

I am going to do that right here, 200 – 140 and that is going to be 60.1062

The probability of A and B are the frequency in this case of A and B is 60.1077

That makes sense, 60 people are here them how many people did Ann know? That is 30 people.1084

The same with Bill because he has invited 100 people and 60 of them knows Ann.1096

When you add all of these up, it makes 140.1101

In a community 80% of households in car insurance or homeowner insurance, 30% carry homeowner’s insurance and 50% in car insurance.1106

If the household is picked at random what is the probability that they are both has an insurance?1123

Once again we could use the general addition rule to figure this out.1128

Because we do not know the probability that the household owns both but we do know the probability that they either own car or home.1133

They at least own one and that is 80%.1152

40% are just home owners insurance and 50% owns just car insurance.1159

We want to know what is the probability that they both have insurance.1171

I will write the general version, it does not matter whether p of c or h comes first – the probability of C and H.1176

I will put in the numbers here.1193

I will subtract these from both sides and that gave me 10%.1197

10% both have insurance.1222

That is for the addition rule.1224

Thanks for using www.educator.com.1227