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Post by Angel Evan on March 28, 2013

Question about probability. Suppose we have a projected audience of 10,000,000 people. Through survey data, we see that 85% of respondents use email on a daily basis. Would it be statistically correct to say that there is 85% probability that the target audience uses email on a daily basis?

Sample Spaces

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 0:07
    • Roadmap
  • Why is Probability Involved in Statistics 0:48
    • Probability
    • Can People Tell the Difference between Cheap and Gourmet Coffee?
  • Taste Test with Coffee Drinkers 3:37
    • If No One can Actually Taste the Difference
    • If Everyone can Actually Taste the Difference
  • Creating a Probability Model 7:09
    • Creating a Probability Model
  • D'Alembert vs. Necker 9:41
    • D'Alembert vs. Necker
  • Problem with D'Alembert's Model 13:29
    • Problem with D'Alembert's Model
  • Covering Entire Sample Space 15:08
    • Fundamental Principle of Counting
  • Where Do Probabilities Come From? 22:54
    • Observed Data, Symmetry, and Subjective Estimates
  • Checking whether Model Matches Real World 24:27
    • Law of Large Numbers
  • Example 1: Law of Large Numbers 27:46
  • Example 2: Possible Outcomes 30:43
  • Example 3: Brands of Coffee and Taste 33:25
  • Example 4: How Many Different Treatments are there? 35:33

Transcription: Sample Spaces

Hi and welcome to www.educator.com.0000

We are going to be talking about sample spaces and start to talk about probability and statistics today.0001

First question is why is probability involved in statistics?0007

We will talk a little bit about some probability fundamentals and then talk about what is a sample space.0013

From there we are going to talk about how to make sure we cover the entire sample space.0020

We would not have any columns.0025

We are going to introduce the fundamental principle of counting.0026

We are going to talk about where the probability come from?0030

how can we check whether our model matches the real world?0035

We might have a probability model but how do we know whether that model is any good?0038

They are going to involve large numbers.0044

okay. First, why the heck are we talking about probability and statistics?0047

Statistics is all about samples from the population and we never really know what that population is like.0053

we have an idea of what that population is like and we call it a model.0061

We call it the model of the population. 0065

We want to know how likely is a particular sample, the empirical data that we have, stuff we actually collect, 0068

how likely is this given a particular model of the world, a theoretical model or a theoretical population.0075

whenever you compute probability you are looking at some subset over the total number of whatever you have.0085

in this case, in probably of statistics we are looking at how likely is a particular experimental outcome over all the different kinds of outcomes that we could potentially have had.0096

we need to have a model of the world to figure out this total.0110

model generated.0116

and that will give us the probability or the likelihood of our experimental outcome. 0120

let us start off with a little example.0128

Just a little case that might be helpful for us to wrap our minds out.0131

I like to drink triple coffee because I just add like sugar and cream but I want to know can people tell the difference between El Chico and expensive coffee?0136

Can we tell between cheap and gourmet?0151

Like all of these expensive brands where they have coffee that is apparently like monkeys eat the coffee bean and cool it out.0154

Sometimes in the stomach make the coffee bean better.0165

It is like most of the expensive coffees to buy.0169

It sounds good but it is supposed to be awesome.0172

Is it really awesome?0176

Can people tell?0179

I was wondering about why is it expensive like wine as well.0180

Can people tell that it is expensive or not?0184

What model of the world you might have is that these people are just guessing.0186

We cannot tell the difference.0193

They are only guessing between gourmet and cheap coffee.0195

When you go to Starbucks and pay, it is nice because they have music and stuff and that coffee just tastes better than Mc Donald’s coffee.0198

Maybe when people actually tastes the coffee they are just guessing.0211

Let us say we want to do a taste test with 100 coffee drinkers and we just give them little cups that looks like the same thing 0216

and one has expensive coffee in it and the other has cheap coffee in it.0229

Let us say we have the n of a 100 people.0235

100 taste testers.0238

Let us say no one can actually tell the difference, does that mean everybody will get it wrong?0239

We will have 0 in a 100 people correct, probably not.0246

Even if they are just guessing, it might be reasonable to expect something like about 50% of people being able to tell the difference.0251

We might say 50 out of 100 correct.0262

That will be pretty reasonable, still reasonable to assume model.0271

Let us say 90 out of 100 people got it correctly.0282

How probably it is hard to do if everyone is just guessing.0292

If it is 90 out of 100 correct then it is difficult to see the model, the model maybe wrong.0297

In that way we can look at the data that we have and see whether our model is likely or not.0322

Let us say everyone can tell the difference, does it have to be 90 out of 100?0335

Could it be 89 out of 100?0345

Could it be 88 out of 100?0347

Could it be 70 out of 100?0356

Could it be 60 out of 100?0358

When we draw a line for people can tell the difference.0360

If we say that earlier 50 out of 100 got it correct, do we say that this model is likely?0365

Here we might say model is more likely than this scenario.0381

If everyone can tell the difference this might be a more reasonable data to see.0395

We do not expect that.0405

In that way it is important to know the probabilities of these different outcomes.0410

How likely is 50 out of 100? 0416

How likely is 89 out of 100?0421

Is one more likely than the other given that particular model of how the world works.0423

In order to create a probability model we want to talk about a couple of different things to help us get better.0428

To help us get on the same page about probability.0439

As we have talked about before, when we talk about probability as an outcome we usually talk about it as P(z).0444

This might be the probability of being correct.0453

We might start with one taster to make it easy.0457

That probability that one taster is correct given that they might be only guessing, maybe let us say 50%.0463

There is only 50% chance that they are correct.0472

The probability of being incorrect might be the other half.0476

Correct and incorrect are what we think of as distinct events.0485

You cannot be correct and incorrect at the same time.0491

It can only be one or the other.0494

We call that mutual exclusivity.0496

Because we have these join events, these probability should add up to 100%.0498

It can only be correct or incorrect.0506

That is the only two zones of this space.0507

We have covered that whole space and it adds up to total probability of 1.0511

That is how probabilities always work.0518

If you have covered the entire sample space, an entire sample space value is 1.0521

How about 2 tasters?0528

What is the probability that both of them guess correctly?0532

It is helpful to think about what is the entire sample space?0538

What are the different scenarios we can have?0545

Number one taste coffee and number two taste coffee, that is the outcome that we are interested in.0548

There is a chance that 1 might get it correct but 2 does not.0556

There is also a chance that 2 gets it correct and 1 does not.0562

There are only different outcomes.0566

It would be helpful if we could figure out the entire sample space and assign probabilities to each zone of that sample space.0569

How do we do that is the question.0576

That is actually a very old question.0581

In the 1700’s there were two mathematicians, one guy is a French guy.0584

They have this argument about what is the probability of having two heads in a row?0593

That is similar to this idea what is the probability of getting 2 correct guesses from our tasters?0607

It is the same problem.0621

It is what we call isomorphic, they are the same structure.0625

The other one was saying they are 3 different situations that you could have.0630

One situation is that you could flip heads, or tails, that is not what we are interested in.0637

We are interested in 2 heads.0647

There is another possibility that the first flip and second flip is heads.0648

This is what we are interested in.0654

There is another possibility that the first flip is heads and the second flip is tails.0657

That is not what we are looking for.0664

In doing this model we have this 3 situations.0668

The first has the number of heads as 0.0674

The second has the number of heads as 2.0677

The third has the number of heads as 1.0683

In each situation has a probability of 1 and if you add them all up you will get 1.0687

He has prepared himself but the other one came along and said I think you left out something.0696

Situations are the first of this test should be symmetrical, that should be equal to all the situations or the first flip of heads.0706

I do not know why you have the first flipping heads are more likely.0717

If you add these 2 together you have 2 heads versus 1 head, what is that more likely than the first flipping tails.0723

That does not make sense. 0731

One goes out that he thinks that the sample space is like this.0732

First flip tails and second flip heads.0736

First flip tails and second flip is also tails.0741

First flip heads and second flip heads.0744

That is what we are interested in.0747

First flip heads and second flip tails.0749

If you look at this, the number of heads that is being 0 in this one has 1 flip probability.0753

1 out of 4.0767

Having 2 heads, that is what we are interested in, that is this one and that is also 1 flip probability.0768

There are 2 different ways where you could have 1 head and the other one being tails.0778

Here is 2 of them and that would be ½.0788

If you add all of these up you will get a total of 1.0792

This is the right probability model not this.0795

I hope you could see there is a problem with there.0803

One issue with this model is he is going to make a complete list of all the different outcomes that he could have.0816

All possible outcomes that is what we mean by the entire sample space.0824

If you have all the possible outcomes in all these different zones.0830

Then we would cover the entire sample space and that is equal to 1.0835

This guy is missing some of the possible outcomes.0839

The other one got it right because he listed all of the possible outcomes that could have happen.0845

The sample space is the complete list of all outcomes.0850

Remember this joint means, another way of saying it is mutually exclusive which means that no joint events can happen at one time simultaneously.0858

You can only have one or the other.0873

All of the outcomes in the sample space must have a total probability equal to 1.0875

Each of these probability or outcomes must have a probability of between 0 and 1.0881

If in some event, like in even A has a probability of 0, this means that there is no chance that this is happening.0890

If we have another event that has a probability of 1 that means it is going to happen 100%.0898

How do we avoid if there is a problem?0907

How do we become like Nicor?0914

How do we make sure that we cover the entire sample space?0915

This is where we are going to involve the what we call the fundamental principle of counting.0919

Before I tell you what that is, I’m just going to show you using what we call an event tree.0923

Let us think about taster 1, he could be correct or incorrect.0930

We think that we have a 50 – 50 probability.0936

This one could be correct or incorrect.0941

Based on that, if taster 1 is correct, taster 2 could be correct or incorrect.0947

But when taster 1 is incorrect, those same events can happen.0955

Taster 1 could be correct or incorrect.0962

There are 4 different outcomes we see whether both correct, taster 1 is correct and 2 is incorrect, 1 is incorrect and 2 is correct, or both incorrect.0966

This is our entire sample space.0981

Presumably each of this in our model where everyone is guessing, each of this has a probability that is equal to each other, ¼.0983

What is the probability that one person gets it right but the other one gets it wrong, we do not care.0997

That would be these 2 added together, ½.1002

Just like the heads and tails case.1006

That is just 2 people, when we have 2 people and 2 different choices, you can think of each like each taster as a slot.1009

A slot where something could happen.1033

Here 2 things could potentially happen.1036

Here another 2 things could potentially happen.1039

If you multiply them together, you will get 4 outcomes.1042

This reminds you of combinations.1047

Those were the same principles because we are looking at how many different kinds of outcomes can we have.1053

That is just for 2 tasters and it already gets a little bit complicated.1062

What about if we have more tasters, for instance 3 tasters?1068

Taster 1 could be correct or incorrect, 2 can be correct or incorrect, 3 can be correct or incorrect.1072

If we sum all these up we have 1 branch here, another branch here, another branch here, another branch here.1099

We have 8 different outcomes.1114

We have C, C, C, we have C, I, I.1117

We know that we have 8 different outcomes, the way that I do it is I write my first one, half of those have to be correct or incorrect.1125

Half of 8 is 4, that is going to be 4 if the taster 1 is correct and 4 where taster 1 is incorrect.1153

Out of these 4, half of them taster 2 has to be correct, taster 2 has to be incorrect.1162

That is the same case for this guy, half of them taster 2 has to be correct, half of them taster 2 has to be incorrect.1171

Taster 3, we know that for each of these cases, because they are identical here, taster 3 has to be correct half of the time and incorrect half of the time.1181

This is a systematically and we sure that each line is different from each other.1202

We have CCC, CCI, CIC, CII.1209

The way you could look at this is you have taster 1, 2, 3, each have 2 possible events and 8 different outcomes.1213

For 4 tasters it would be complicated to draw a tree.1227

Instead I am going to just find how many outcomes we have.1231

Here I have 4 tasters, each has 2 possible events being correct or incorrect.1238

That is 16 possible outcomes.1248

I can use this method where I might have half of 16 is 8, CCCC.1255

I do not have space for this.1266

Maybe I will try to draw it a little bit smaller.1269

CCCC, here is 8 I.1276

I will draw the next one with blue, the other half of these taster 2 has to be correct and half of them taster 2 has to be incorrect.1286

That is going to be 4.1305

Taster 3 half of the time has to be correct and half of the time has to be incorrect.1314

Finally, I will go back to red and we just alternate.1333

I remember having to do this for logic classes.1341

Hopefully your instructors would not ask you to do more than 4.1348

It can be done, you just have to keep track of half of it have to be correct and half is incorrect.1356

This is our 16 sample space and each of them have a probability of 1 out of 16.1365

That is where probability comes from.1373

One is that it comes from observed data, we look at actual data in the world in order to figure out the probability.1379

In fact you might think that it is a 50-50 chance of having boy and girl but actually it is 51% chance of having a boy versus a girl.1386

Those probability might be affected of other things like, in other countries.1398

The second thing is symmetry.1405

Heads and tails are good example of symmetry.1407

There are more reason of thinking of flipping heads is more likely than flipping tails.1414

Whenever you have somebody who is guessing, guessing on a multiple choice test that involves symmetry.1419

What we mean by symmetry is not necessarily but they are the same for each option given that there is no reason the other one is better than the other.1426

The final thing is subjective estimates.1436

This one is how lucky are you to do a get a good grade in this class.1439

No one can actually tell you for sure, you just have a feeling maybe this percent or this percent.1449

Those are subjective not based on hard data.1459

Since we are in probabilities come from, the question that arises is if we have a probability model of the world, how do we know that they are model or theory of the world matches the real world?1467

It will be useless to have a model that is inaccurate that it does not match the real world.1487

Here is where we involve the large raw numbers.1494

What we assume is a reasonable fit to the real situation is we assume that when we can compare the probabilities derive from the model with the probabilities observed from the data.1499

If we have a lot of observed data and that matches with our model, then we would assume it is a reasonable fit.1515

That is what we mean by the raw large numbers.1534

The more data we have the more we trust in that match.1537

If we have match but we have a real small data set then we would not trust it.1545

The larger and larger our data set becomes then if it matches it is pretty good.1552

In this model and Nichor’s model, they predicted different probabilities for getting heads.1558

Flipping 0 head that is 1/3, 1/3, and 1/3 and they all add up to 100% or 1 probability.1569

In Nichor’s model, he thought that this have a 1 heads probability, that a probability of just one heads is ½.1581

If we fit 3,000 coins or you did it in a computer simulation you might get data that look something like this.1592

782 out of 3,000 came out with 0 heads.1604

I should say pairs of coin flips.1612

725 came out with 2 heads in a row.1620

About1500 came out with 1 head and the other being tails.1626

When you look at the probabilities, you just take this number and divide by the total.1635

You see that when we get these particular values, do these match the Nichor model or do these match the other model?1640

It is easy to see that these actually match the Nichor model.1650

Using the large numbers we could say the Nichor model it fits more with the real world than the other model.1655

Let us go into some examples.1665

Which of these statements accurately applies in large numbers?1670

We are looking at the fit between our data and the real world.1674

Does it really predict or look like the real world?1681

An opinion pollster says all you need to do to ensure the accuracy of the poll result is to make sure you have a large sample.1685

That sounds reasonable because we want to make sure that our poll results, if we say who do you think will win the next election?1695

We want to make sure that matches the actual population of voters, if you have a real large sample that is more likely going to match the real world.1704

A casino operator says all I need to do to ensure the house will win most of the time is to keep a large number of people coming to my casino.1713

Let us think about this one.1729

The raw large numbers is about having a lot of data then whatever your data says you know that will probably match the real world.1731

The house winning those are probabilities that are set by the games.1741

How do the games are set up?1749

Having a large number of people coming in affect those probabilities?1752

No, you just have to change those probabilities first.1757

This one is a no.1761

The number of people coming in are not going to have change those probabilities to help the house win more.1763

That is not going to change the probability.1771

The world large numbers does not say that having more data will change the probabilities, 1773

it just says that having more data will help you know what the real world probabilities are.1780

It just helps you understand.1787

It does not help you change the real world.1789

A manufacturer says all I need to do to keep my proportion of defective items is low is to manufacture a lot of light valves.1791

This affects the understanding of proportion.1801

Proportion is percentage and that is relative.1805

If you have a crappy factory and 25% of the valves are defective, whether you have a small number of valves 1811

or large number of valves they are still 25% that are defective.1822

If you have a lot of valves it will not change the proportion.1826

Once again it is wrong, because the raw large numbers does not have a change of the real world probability, 1831

it only helps you understand that or know what they are.1837

Example 2, suppose you slipped a tera coin 7 times, how many possible outcomes are there?1842

Thankfully it does not say list all of them, it just says how many possible outcomes.1852

Think of each coin flip as a slot where one of two things can happen, heads or tails.1857

There are 2 possibilities for each of these.1864

v27 that is our answer.1871

Suppose you roll a dice 9 times, how many possible outcomes are there?1876

It is like to think of each roll the die as a potential event that has 6 different possibilities.1884

Each has 6 and so this would be 69.1898

The other way that you will see the fundamental rule of counting is that it will usually say if you have n possibilities 1910

and k number of events, total outcome, is n^k.1948

Here you could say if you have n possibilities, 6 possibilities for each k events then it is 69.1971

Same thing here, I always forget which is which.1983

This is that idea.1990

You could see it more readily when you see each event as a slot to be filled with possibility.1993

Example 3, supposed 5 taste testers are comparing 3 brands of coffee.2003

What is the sample and all possible outcomes?2012

Here maybe they have tastes one coffee then they have to pick whether it is Starbucks, Mc Donald’s, or Dunkin Donuts.2016

This question is actually a bit weird because it is a little bit big.2027

Let us say that is what this question is asking.2032

What are the possible outcomes?2035

What might these people guess?2042

If I have 5 taste testers and each of them can have 1 out of 3 guesses, Starbucks, Mc Donald’s, or Dunkin Donuts.2044

That is 35.2058

What you want to do is make sure that all of the sample space is covered.2068

If 5 taste testers, you want to have the equal probability of the first one picking Starbucks.2073

The second and third one picking Starbucks.2086

It might be helpful to figure out what his actually is.2094

9 × 9 × 3 = 81 × 3 = that is a lot of possible outcomes.2097

I will just leave it up like that.2115

That is a lot of possible outcomes but usually they would not ask you to draw that out.2121

Example 4, assume the different treatments for anxiety randomly signs each new patients to 1 to 2 levels of exercise and 5 different types of medication.2130

How many different treatments are there?2148

Show the sample space in a tree diagram and as a table.2150

First thing is how many different treatments are there?2153

The first slot will be levels of exercise.2159

They get 1 of 2 levels of exercise.2163

The second slot is 5 different types of medication.2165

I will just call these ABCDE.2169

There are 10 different treatments.2173

Let us get started.2178

First, we will have the exercise and then we will have the medication part of the tree.2181

The exercise part of the tree will be mild and moderate.2188

Medication will be ABCDE.2194

If we look at all the outcomes, the table we could look at it as mild, mild, mild, mild, ABCDE.2207

Same principle as before.2230

Each of these different treatments are equally likely or we wanted to be equally likely in our sample.2232

For instance we look at this treatment group, this group of people or group of experimental cases gets mild exercise they also get medication B.2248

That is the end of sample spaces.2268

Thank you for using www.educator.com.2271