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Lecture Comments (3)

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Post by kate marcus on December 3, 2012

Also forgot to get the square root for standard deviation. 2.34 is what npq is.

0 answers

Post by kate marcus on December 3, 2012

Agree on the need to correct that. The answer is .0464.

0 answers

Post by James Ulatowski on December 31, 2011

Example 3 Error, the P(X=0) is .0464 not .0427, you raised the (.4) factor to power of 1 instead of 0. You correctly said it would equal 1, but then made the exponent 1 instead of 0. Being a teacher myself, I know how easy it is to get ahead of yourself and make a "whoops" error. A little arm waving here and there in general discussions, but the worked examples allow me to figure out what the key points are - so, eventually clear.

Binomial Distribution

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
  • Discrete Probability Distributions 1:42
    • Discrete Probability Distributions
  • Binomial Distribution 2:36
    • Binomial Distribution
  • Multiplicative Rule Review 6:54
    • Multiplicative Rule Review
  • How Many Outcomes with k 'Successes' 10:23
    • Adults and Bachelor's Degree: Manual List of Outcomes
  • P (X=k) 19:37
    • Putting Together # of Outcomes with the Multiplicative Rule
  • Expected Value and Standard Deviation in a Binomial Distribution 25:22
    • Expected Value and Standard Deviation in a Binomial Distribution
  • Example 1: Coin Toss 33:42
  • Example 2: College Graduates 38:03
  • Example 3: Types of Blood and Probability 45:39
  • Example 4: Expected Number and Standard Deviation 51:11

Transcription: Binomial Distribution

Welcome to 0000

Today we are to be talking about binomial distribution.0001

So far we been talking about discrete probability distribution these models of what the probability space of all the samples, of all the different outcomes look like.0004

Now the binomial distribution is the special case of these discrete probability distributions and binomial is whatever two.0019

We are to be talking about this special case because it is actually ends up being one of the most frequent probability distributions that you end up using.0029

In binomial distributions which are particularly interested in this 1 out of 2 outcomes like heads or tails or black or white.0039

You know 1 out of 2 things right and how many successes which we call K number of successes and trials.0052

Something like how many heads out of 10 tosses.0059

Then we will do a quick review of the multiplicative rule because that can be important for dealing with discrete probability distribution in a binomial way.0062

How many outcomes with case successes there is going to be a formula that you can use to figure this out real quickly.0077

Then we are going to talk about probabilities in a binomial distribution. 0085

How do we actually get the spread of all those probabilities?0090

In such a distribution of how do we find expected value, and how do we find the standard deviation?0093

Discrete probability distributions as you know these deal with outcomes that are nameable and countable.0102

There are an infinite number of possible outcomes. 0113

There are a discrete finite number of them. 0118

They are simple space is all the outcomes, just flat out all the outcomes but the probability distribution is taking a sample space and also finding 0122

the corresponding probabilities but frequently one of the most frequent probability distributions, you will run across is the binomial distribution. 0134

That is really going to be familiar with this particular one and all of its all of its quirks and foibles. 0147

Let us talk about the binomial distribution.0155

Many random variables they are really counting the number of successes in an independent trial.0160

By successes, we do not necessarily mean that it have to be like a winning trial or anything.0168

It just means whatever you are interested in right whatever outcome out of two outcomes you are interested in, 0174

a lot of times the random variable is counting how many times that interesting event happened in a number of trials.0180

Let us say we have 20 independent trials we could have 0 successes 1, 2, 3, 4, 5 all the way up to 20.0188

The number of successes will be our K value and K can range from 0 to 20.0197

Just to give you examples of some it might be counting the number of heads in a random sample of 10 flips of the coin. 0203

Here the random variable x is a number of heads.0213

Another example, if counting the number of children who been diagnosed as autistic in a random sample of 1000 children. 0219

Here, in that case, the random variable x is number of children and diagnosed with autism. 0228

Another example, if counting the number of defective items in a sample of 20 items notice that you know when you think of the word defective, 0239

you do not really think of that as successes, but it is really what we are doing is we are counting some outcome of interest for every n trials.0257

In this case n is 20.0263

Here x equal number of defective items and just to round it out let us talk about what the n is here.0267

n in this case is 10 and in this case is 1000 and n in this case is 20.0286

Let us think about why these are called binomial situations.0298

In each of these situations you either have a success, the events of interest or you have a failure.0304

It is not an event of interest.0315

Here what we would see is in all of these different situations there are two outcomes that you have.0318

You can either have a head or tail, and they both have some probability and those probabilities add up to 1.0330

It has to because you only have those two choices. 0338

There is here it is either being diagnosed as the probability of being diagnosed of autism and the probability of not getting a diagnosis. 0341

There is only those two outcomes.0348

Here it is either being defective or not being defective.0352

It is one or those two outcomes.0357

These are binomial situations because there is 2 outcomes that are disjoint and so it is 1 or the other, right.0360

If you add the probabilities of those outcomes, if you add the probability of outcome 1 + the probability of not outcome 1, the other one, then you should get 1.0373

Another way to put it is the probability of one outcome is equal to 1 - the probability of not that outcome. 0394

That is the other way you can think about this.0409

Let us briefly review the multiplicative rule.0413

Remember, when you had to think about things like this where the proportion of adults in the US with at least a bachelor's degree is 29%.0419

Suppose you picked for adults at random what is the probability 0430

that exactly 2 have a bachelor's degree and some of the things that we did in order to find these probabilities0433

is we imagine having slots for these different adults and you either have the probability of getting the Masters degree.0439

Let us say these 2 get the 29 or you have the probability of not having the Masters degrees, the other outcomes that would be flip side of the 1-.29 so that would be 71.0451

This is one combination but this would be that they would have bachelor, bachelor, no bachelor, no bachelor.0469

Then there are other combination of exactly 2 having a bachelor's degree and so you have to find those other combinations as well as b and b.0480

Just to recap the multiplicative rule in order to find this particular guys outcome we would have to multiply these probabilities together.0501

I want you notice something.0516

We are not actually going to do this probably we have done it before, but I want you to notice that let us say we wanted 0518

to find this the probability of getting this particular outcome, although the order will change in multiplication it does not really matter what order it is.0524

The probability of this outcome is exactly equal to the probability of this outcome.0537

That can be important for us to keep in mind and I just like you do realize that we are multiplying these probabilities together.0552

Just to remind ourselves a little bit more about the multiplicative rule now and it is much more likely that adults that if you pick the random adults, 0563

they would not have bachelor's degree.0575

Which of these combinations is more likely and all 4 people having bachelor’s degrees or none of them having bachelor’s degrees. 0579

If we just think about all the fact that you know only 29% of adults in the US have at least a bachelor's degree 0592

You are going to know that this is the combination of all 4 having a bachelor's degree is much less likely than all of them not having a bachelor's degree.0600

That make sense here and these probabilities are witness to that.0618

Before we use the multiplicative rule it is going to be handy for us to know is exactly how many outcomes with case successes we will find.0622

We will be looking at relatively small sample spaces.0633

Maybe out of three coin tosses how many have exactly 2 heads?0637

In those kind of cases we can actually list out all the possible outcomes and just count how many of these outcomes have only two heads.0643

As soon as the sample space get a little bit bigger like 6 tosses, 7 tosses, that is 26, 27 0651

and you know they do not call it growing exponentially for nothing like those numbers get really big and fast.0660

If you think about 10 coin tosses and how many of those outcomes have exactly 2 heads that is going to be way impossible for us to actually manually draw out.0667

There is a shortcut, but before I teach you the shortcut I want you to see how it fit together with the manual way of doing it.0682

I’m going to use this example, the bachelor's degree and here is what you are going to see.0692

Here we have the manual lists of outcomes and so here I have to draw person number 1, 2, 3, 4.0699

I will draw another list for 1, 2, 3, 4.0723

It is just that we know we need it is going to be 24 and that 16 and I am going to draw 8 in each column.0727

Here we are going to start off with half of these outcomes. 0737

The first person has a bachelor's degree, half of these outcomes the first person does not have a bachelor's degree. 0749

Half of these the second person has a bachelor's degree and half of these as well the second person has a bachelor's degree. 0758

It gets to be quite a bit.0777

Here we go, that is our entire sample space of all the different outcomes. 0792

Now each of these outcomes are not equally probable, it is not like heads and tails.0820

We know that this outcome is much less probable than this one. 0825

This one is also much less probable than this one.0830

We know that they are not only even but this is at least the list of all the possible outcomes.0833

We want to know what is the probability that exactly 3 will have a bachelor's degree?0839

It has to know how many of these outcomes have at least three that have a bachelor's degree.0844

This one, this one, this one, this one, and that is it.0851

4 out of 16 of the outcomes have at least three people who have a bachelor's degree. 0865

Now this does not mean that the probability that exactly 3 have a bachelor's degree is 4 out of 16 because each of these are not equally probable.0877

It is good to know how many of them there are.0889

There is a shortcut to get this number, this number of 4 how many outcomes with K successes and here K is 3 and n is 4 out of 4 adults.0893

Let us put back here n is total number of trials or total number of spots, number of independent trials and K is number of successes generally.0910

In this case K is number of bachelor's degree holders and n is for adults. 0936

We can actually use an insight from permutation combinations like probably long time ago for most of you in order to find this number 4.0945

In fact we could use n2k, this is also written as sort of like these big parentheses nK and there is also another way you could write it words like n and c, 0960

either for choose a combination not especially sure which one.0969

Here is the actual formula for it.0978

If you have n choose K that is how you say that and then what you really want is how many relevant combinations0983

can you have where you have any number of slots but you have K number of successes for those n slots.0992

This is going to be n factorial that is like if you have three factorial that will be 3 × 2 × 1 over I always 1000

remember this picture first n - K factorial and the reason for that is that you will end up having this.1009

If you have like 4 factorial and you have 4 – 2 let us say that will be 2 factorial that means you will start listing the factorial up until K.1020

It is like here is our 4 slots and it could be 4 3 2 1, but you will only do the factorial up until the number of successful slots.1042

And also K down here on the bottom that would be k is 2.1057

This is the formulae that will give us this nice number of 4 combinations having three successes out of 4.1066

Let us see if this works, at least for our example.1078

In our example n is 4 so that would be 4 factorial.1082

Let me just erase this stuff down here we do not really need it.1088

That will be 4 factorial over n - K which is 4-3, which in the being 1 and K is 3.1094

Oftentimes I advise people like on the SAT and stuff like you do not want to actually calculate out the factorial always 1110

because sometimes you just can cancel without having to actually calculated.1120

This one is the factorial 1 or 1 is just 1 but we could just forget that.1125

4 factorial or 3 factorial it is 4 × 3 × 2 × 1 / 3 × 2 × 1.1130

I could just cross out the 3 × 2 × 1 which was actually 4.1141

I do not have to multiply anything and guess what we got, 4.1145

The nice thing about this boring life that you can use that when you have an inordinately high number of independent trials1152

you do not have to actually pretend to write out 10 slots in all the different combinations 1161

and you can actually just put it into the formula and I will tell you how many outcomes with K successes there are.1168

Once we know that now we need to put together the multiplicative rule with the number of outcomes that we learn.1176

Here is the n choose k stuff that we learn and the multiplicative rule helps that calculate the probability of one particular outcome.1189

You want to put those together and I would introduce a slightly different notation system 1200

before we look at the probability of some events the probability of an occurring.1211

It is the same thing, except here is the likelihood x is our random variable and any binomial distribution we actually already know what x.1221

It is not just a random variable X is actually the number of successes.1240

We actually already made up a letter to symbolize number of success that is called K.1256

x = K.1266

K can be all sorts of discrete number straight like 123456 however many trials you have K is that many + 0 number of success.1270

What we are really looking for is all the different probabilities where X = k and K can have a range. 1285

That is our binomial distribution.1295

The set of all the probabilities where x=0, x=1, x=2, x=3.1300

All those probabilities altogether, that set makes up the probability distribution that we call the binomial distribution. 1309

This is what we are looking for.1317

now, in order to find this we have to know the probability of getting that particular outcome and that is actually quite simple 1320

because we talked about the example where we have 4 slots and 2 of them have bachelor's degree.1332

That would be BB and n or and nn BB or B and nB.1343

They all have exactly 2 people have bachelor's degrees and we know how to find it. 1355

They all have the same probability.1362

How can we express this in a more abstract form?1365

We understand that concretely how can we express it in an abstract form?1373

There is a straightforward way of doing so.1379

Consider that P is the probability of the K happening, whatever success rate. 1383

So the probability of success we are going to call it p for now just to shorten down the notation.1391

How many P do we have?1405

We have k number of P.1407

It is p^k.1411

In the case it is .29 probability of the success and K is 2.1414


That accounts for this part. 1426

How do we count for this part?1429

Well that is 1 - p because we have to account for the non-successes and how many of those non-successes do we have?1431

We have n - K and it has 2 people out of 10 we would have 8 other slots filled by non-success.1441

In this case we have 4 slots and 2 successes so how many of slots are filled by non-successes?1454


This will give us the probability of exactly 1 of these combinations.1466

Remember there is a whole bunch of the different combinations if we know how to get that number .1472

You multiply all of that by the number of different combinations that you can have and that is n 2k.1480

If this probability they are all here, they are all the same, that times however many of those outcomes that you have.1493

That is the probability where x=k.1502

You can plug in numbers for k or you do not have to plug in numbers for k but there you go expected value in stdev in a binomial distribution.1513

Once again this binomial distribution then we will have something that looks like this.1523

Like you can put it in a table or in a histogram but for now I will do it like this.1533

Out of 4 results number with bachelor's degree.1540

You have 0, 1, 2, 3, up to 4.1551

Then you want the probability of this outcome.1557

The probability where x=k and these are all our k.1562

K=0, 1, 2, 3.1573

We express this as p(x=0), p(x=1), so on.1576

You could express this as a table chart just like all of our probability distribution.1584

How do we find the expected of value of this probability distribution?1594

We know how to find these.1602

We have our formula that we have just learned and we could also reason it out.1605

We want to know how many combination have x=0.1613

0 number of successes.1617

Then we want to multiply that by the probability of those successes.1620

Here we would have all of this probabilities and we would have all these k or x=k.1624

Then we want to know what is the expected value?1637

What is on average basis, an average of them together?1641

On average what would be the number of bachelor's degrees I would expect when I sample all the 4 results independently in populations?1646

Before we had expected value of and another way of writing is mu^x.1655

Before we had to do all this crazy multiplying thing but now we could think of it as n × p.1670

P being the probability of success of whatever your success is and here this is the bachelor's degree.1684

In this case it would be n = 4 × .29.1693

That is the expected value of this distribution.1699

If you use a calculator to do this, on average what is the k value on average?1712

4 × .29 and here it says 1.16.1732

Let us think about this.1740

That say that on average you will have 1 being the most frequent number of adults out of 4 that will have bachelor's degree.1744

That makes sense because it is not a super likely scenario.1761

There is a .29% chance and in some sense if you think about that, that is close to ¼ like 25% chance.1766

It makes sense that out of 4 people how many are you going to expect to have bachelor's degree?1776

It is going to be 4 × .29 it is going to be ¼ of n.1785

It is going to be 29% of n.1793

In that way this number ends up making sense here.1797

What about standard deviation?1803

Previously we have talked about how to write this when we talk about the expected value .1807

Remember expected value means it is not the mean of the population.1817

It is not the mean of the sample.1822

It is the mean of the probability distributions.1824

Here this standard deviation of the probability distribution how this spread around this 1.16 value?1829

How this spread?1838

Here it might help to get the variance first the we will just square root this to get the stdev.1840

In order write that it is sigma2 but with the sub x down here to indicate that it is an expected value.1854

Here we have n × p × 1 – p.1863

We have to account for successes.1871

We have to account for failures.1873

You have to account for how many slots there are.1875

Square root that whole thing to get n × p × n-1.1877

That is the standard deviation of a binomial distribution.1885

These are specific forms of the general form.1895

You can always use the regular expected value in stdev that you would normally use.1901

Multiplying across and adding them up.1910

These are some short cuts that work because when we are talking about binomial distribution each slot has fixed probability of p, 4 successes.1915

Because of that it puts down some of our work.1929

One other thing to know about this is as n increases, as the sample size of n increases the binomial distribution get more and more normal.1932

Think about it, at the very extreme we are interested in the population of n of the US.1954

We have in our sample N-1, that is n.1965

If that is how big our sample is, that is almost like having everybody in the US.1972

Basically, as n gets bigger you will end approximating a normal distribution in the binomial distribution.1979

That is helpful because it does not mean that our population is necessarily abnormal.1996

It is just means that if we have the probability distribution that becomes more normal.2004

This principle will become even more clear when we talk about the sampling distribution of sampling.2009

We will get more into that later but I just want to throw that in there.2017

Let us move on to examples. 2021

Example 1, in an all day tennis tournament each round of the competition will begin with a coin toss on the 4 different courts to determine who will serve.2026

In any 1 round what is the probability that exactly 2 people will call their respective coin toss correctly?2035

If you think about this, you have 4 courts and 4 coin tosses and we are looking for the probability that exactly 2 people will fall their respective coin tosses correctly.2042

2 people will be correct and 2 people will be incorrect.2058

This is not the only one.2063

There are many different combinations.2065

I’m not going to deal with all of those combinations even though I could.2073

I would not use my n2k combinations idea in order to figure out how many outcomes will I expect this.2077

In this case n is 4 but 2 is the number of successes.2094

That is going to be 4^/4 – 2^.2105

I know that this and this will be just 4 × 3 and then I can cross out 2 ×.2117

I will put this in here 2 × 1.2125

I do not need the 1.2129

6 of my combinations will have exactly 2 people call their coin tosses correctly.2130

Given that one thing that is nice about this is that all of these probabilities are exactly the same because the probability of being correct is .5 2144

and the probability of being incorrect is also .5.2154

Just to illustrate for you I am going to put in to 1 – p form so that you could see.2160

Here I want to know the probability that x my random variable = 2 successes.2167

In order to find that I put in my number of outcomes that have exactly 2 successes × the probability of success which is .5 × k.2175

1 - .5 the probability of being incorrect × n –k.2198

These two are the same.2209

Let us simplify this.2210

I know that this is 6 × .52 × .52.2213

I know I could put this together and just put 6.54.2226

Let me just get my Excel calculator here.2241

6 × .54 and I will get .375.2253

My probability of getting exactly 2 people calling their respective coin tosses correctly is 37.5%.2265

Example 2, given that 29% of the population of adult in the US who have a bachelor's degree or higher,2282

create a probability table for the number of college graduate for any group of 7 randomly selected adults.2290

What is the probability that given the sample have 5 or more college graduate?2298

What this is asking for is the probability table that looks something like this.2303

Here we have k number of bachelor's degree table and that would be 0, 1, 2, 3, 4, 5, 6, 7.2310

We also want to know where the probability where x = k.2325

We could just find out those formula.2329

Just to give you an idea I will show you the first one.2340

We could do probability of x = 0, 0 number of successes.2344

That is going to be n2k.2352

N is 7, k is 0.2354

Just to remind you 0 factorial is just 1 not 0 × the probability of success which is 290 and 1 - .29^n -0 which is 7.2359

Let us look at choose and see what choose means.2387

Unfortunately choose means literally juvenile.2408

This is not what we want.2422

It is useful to try combinations.2425

Let us see what this one says the number of items like n and the number of items that each combination which is k.2436

This is exactly what we are looking for.2449

N^/k^ × n – k^.2451

We want to choose for a number we want to put n, for number chosen we want to put k.2458

We can use combine.2466

It is great.2467

Before I do that I am going to create a little table for myself so that I can see things.2469

Here is 0, 1, 2, 3, 4, 5, 6, 7 so that they do not have to put in my formula again and again I can just copy and paste.2480

P where x is = k so this k is going to be combine 7, 0, 7.2489

I am going to make a formula so I choose that 0 × .290.2504

Excel know order of operations so it is going to do the power before multiplying.2515

1 - .297, 7 will always stay the same that is why I am just checking it in, - k^.2524

The probability of having 0 people have bachelor's degree is 9%.2551

I am just going to copy and paste that all the way down.2559

We could see that 9% might look small but that is larger than all of 7 actually having bachelor's degree.2565

We are looking for what is the probability that a sample will have 5 or more college graduate.2575

Here we can use the addition rule to put these 3 probabilities together.2585

Just to show you.2591

I will just write the 5 and 6 probabilities.2595

This is .0217.2607

Here you put all of this and the rest but what is the probability that the sample will have 5 or more college graduates?2624

We could put together what is the probability that x is > or = to 5 or more?2633

That will be the probability where x is =5 + the probability where x=6 + the probability where x =7.2649

I will bring this back if I just add this up.2667

The probability where x is > or = to 5 + the sum of the three and we get .0248.2673

That means a chance of 2 ½ % chance of randomly selected 7 adults and finding that at least 5 of them have a college degree.2695

Although it seems like it will take a long time that is why I am just showing it to you I felt you could write this one for each of these rows.2715

Excel comes in handy.2735

40% of blood donors have type A blood, The blood bank need 2 type A donors to walk in 2738

and the blood bank will test 10 random blood donors and count the number with type A blood.2749

If they say calculate the number with something what is the other?2755

Then you will know binomial distributions.2760

What is the probability that the blood bank has fewer than 2 type A donors?2766

If they have said what is the probability that blood bank has 2 type A donors versus type B donors this one will be a binomial distribution.2771

It could be type A, B, AB, or O but this is just the same what is the probability that they are A or not A?2783

That is how you will know if it is a binomial distribution.2792

What is the probability that the blood bank has fewer than 2 type A donors?2795

It is nice to just start off with this idea that there is going to be 10 donors and 2 of them need to have type A blood.2800

That would be the probability of anyone of these combinations would be .40, that is p^ k × 1 – .4.2814

60% could not be A probability × 210 -2.2831

That is the rest of the other 8 slots.2845

We need to know how many of these combinations we have so that would be n2k which will be 10.2848

This will give us the probability where x=2.2859

Is that what is this asking?2865

No, that is not.2868

This is not good enough.2870

What we need to know is the probability where x= fewer than type A donor.2871

These are the situations that they do not want.2887

How do we get this?2892

That is going to be the probability where x = 0 + the probability where x = 1.2895

We combine that.2905

If this is x = 2 we could obviously do this for x = 0 or x =1.2908

This would be 100, .400, .6 the not A probability10.2920

10 choose 11,.609 that is the rest of the slots.2937

I could just use my handy Excel function now that I know combine I will put 10 choose 0 × .40 which is 1 × .610.2955

That is the probability of getting 0 out of 10.2994

It is pretty low. 3001

It is less than 1% chance.3003

.24% chance.3005

We are not in danger for that happening.3008

Let us look at the probability of only 1% having type A blood walking in.3013

That × .41 and then .69 and that is a 4% chance.3019

If we add these up what do we get is still less than 4%.3036

This would be =.0427.3047

A little more than 4% chance that the blood bank will get fewer than type A donors walk in.3059

Example 4, 2.4% of students in a large state university consider themselves multiracial.3070

In a random sample of 100 students what is the expected number of multiracial students.3080

What is the standard deviation?3087

This is a good one because this is definitely a case where you cannot imagine even with 10 blood donors.3089

100 for sure is we cannot solve way too much of your life writing all the different combinations.3098

This a good example of situations that you run into where you are going to need this binomial distribution ideas. 3105

What is the expected number of multiracial students?3117

We could make this giant probability distribution of 0 to 100 to the expected value or we know that there are some regularities to the expected value in a binomial situation.3122

We could also write it as mu sub x and we know that this is n × the probability of success.3146

Here success is being multiracial.3166

Our n is 100 and what proportion of those students will be multiracial?3170

Just 10 × the probability of success × .024% that is 2.4.3181

The expected number of multiracial students around 2.4 and that makes sense.3196

What about standard deviation?3205

That would look like this will be sub x.3207

I would like to start with the giant square root to remind myself where I am going.3216

You could put it at the end but sometimes I forget to put it in.3225

N × p of being multiracial and you also have to count for probability of not being multiracial.3230

That would be 100 × .024 × the other side of that, that is the 97.6% of not being multiracial.3240

That is 2.4 × .976.3267

I am just going to use my handy calculator.3272

2.4 × .976 = 2.34.3278

The nice thing about standard deviation is that it is always in the same unit as the mu this 2.4%.3289

The spread is quite small having given that it is 100 students.3304

That is it for binomial distribution, thank you for using 3310