For more information, please see full course syllabus of Statistics

For more information, please see full course syllabus of Statistics

### 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
- Roadmap
- Discrete Probability Distributions
- Binomial Distribution
- Multiplicative Rule Review
- How Many Outcomes with k 'Successes'
- P (X=k)
- Expected Value and Standard Deviation in a Binomial Distribution
- Example 1: Coin Toss
- Example 2: College Graduates
- Example 3: Types of Blood and Probability
- Example 4: Expected Number and Standard Deviation

- 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

### General Statistics Online Course

### Transcription: Binomial Distribution

*Welcome to www.educator.com.*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 probabilities*0433

*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 2 ^{6}, 2^{7}*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 2 ^{4} 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 combinations*0983

*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 trials*1152

*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

*P^k.*1421

*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

*4-2.*1463

*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 sigma ^{2} 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 × .5 ^{2} × .5^{2}.*2213

*I know I could put this together and just put 6.5 ^{4}.*2226

*Let me just get my Excel calculator here.*2241

*6 × .5 ^{4} 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 29 ^{0} 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 × .29 ^{0}.*2504

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

*1 - .29 ^{7}, 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 × 2 ^{10} -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 10 ^{0}, .40^{0}, .6 the not A probability^{10}.*2920

*10 choose 1 ^{1},.60^{9} 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 × .4 ^{0} which is 1 × .6^{10}.*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 × .4 ^{1} and then .6^{9} 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 www.educator.com.*3310

0 answers

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.