For more information, please see full course syllabus of Probability

For more information, please see full course syllabus of Probability

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### Markov's Inequality

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
- Markov's Inequality 0:25
- Markov's Inequality: Definition & Condition
- Markov's Inequality: Equation
- Markov's Inequality: Reverse Equation
- Example I: Money 4:11
- Example II: Rental Car 9:23
- Example III: Probability of an Earthquake 12:22
- Example IV: Defective Laptops 16:52
- Example V: Cans of Tuna 21:06

### Introduction to Probability Online Course

### Transcription: Markov's Inequality

*Hi and welcome back to the probability lectures here on www.educator.com, my name is Will Murray.*0000

*Today's lecture is on Markov’s inequality.*0005

*Markov’s inequality is one of two inequalities that you can use to estimate probabilities quickly,*0008

*the other one was Tchebysheff's inequality.*0015

*The next lecture will be on Tchebysheff's inequality.*0017

*If you are looking for that one, just skip ahead on the next video.*0020

*If you are looking for Markov’s inequality, here we go.*0023

*Markov’s inequality is a quick way of estimating probabilities based only on the mean of a random variable.*0027

*You all you have to know is the mean of a random variable or the expected value.*0033

*Remember that mean and expected value mean exactly the same thing.*0038

*One important condition that you need to use Markov’s inequality is that, your random variable has only positive values.*0042

*You have to be estimating things that can only be counted in positive numbers, like a number of customer at a business,*0052

*or the number of miles a car is driven, or the amount of money you have assuming you are not allowed to go into debt,*0059

*things like that, that can be only measured using positive values.*0065

*If it is a random variable that takes on negative values then Markov’s inequality is not necessarily true.*0069

*The way it works is you have some constant number, that is this value A here.*0076

*What we are going to do is estimate the probability that the variable will be bigger than that value A.*0082

*Markov’s inequality gives you an answer for that.*0089

*It says that the probability is less than the expected value of Y or that is the same as the mean divided by A.*0092

*Maybe another form you might have seen this inequality is μ/A*0101

*because remember that people use μ as the expected value as a shorthand for the expected value.*0106

*There is one thing I want to emphasize about Markov’s inequality which is that it is really a one sided estimation.*0114

*This is a one sided bound, it gives you an upper bound.*0123

*It does not tell you that the probability is equal to that, it just gives you an upper bound, one sided bound on the probability.*0132

*Whenever you answer a question using Markov's inequality,*0146

*your answer will always be something like the probability is less than something or the probability is greater than something.*0151

*You can never say that the probability is exactly equal to something based on Markov’s inequality.*0157

*It just gives you a one sided upper bound.*0163

*You can reverse this, we said we are calculating the probability that Y is greater than or equal to A to the basic form of Markov’s inequality.*0167

*You can also switch that around and ask what the probability of Y being less than A?*0179

*That is exactly the opposite of Y being greater than or equal to A, we get the complement of that.*0185

*The probability is now greater than or equal to 1 - the expected value of Y/A.*0192

*Remember, we can also write that as the expected value of Y as μ.*0200

*You can say it is greater than or equal to 1 - μ/A.*0205

*Again, this is a one sided thing.*0210

*It can tell you that the probability that Y is greater than A is less than something.*0213

*It can tell you that the probability that Y is less than A is greater than something, but you can never reverse those.*0218

*You have to be very careful about how you use Markov’s inequality which way into inequality go,*0225

*more practices as we go through some of the examples.*0231

*You also have to be careful never to say this probability is equal to something*0234

*because Markov’s inequality will never tell you that.*0239

*It will always just say this probability is less than something or this probability is greater than something.*0241

*Let us check this out with some examples.*0248

*First example, have we done a survey on a particular college campus and apparently,*0252

*the students on this campus are all going to have some cash in their wallets.*0259

*It turns out that the average amount of cash that these students are carrying is $20.00.*0263

*The question is, if we meet a student at random and ask here how much cash she is caring,*0270

*what is the chance that it should be carrying more than $100?*0275

*Let us estimate the chance that she is carrying less than $80.00.*0278

*Let me first emphasize that this is a situation in which Markov’s inequality does apply*0282

*because the amount of cash students are carrying is always going to be a positive amount.*0291

*The smallest you can be caring is 0.*0296

*You could be carrying a definite amount of cash.*0299

*You could be carrying thousands of dollars in cash but you cannot be carrying a negative amount of cash.*0305

*This is always the amount of cash that you are carrying is always positive.*0311

*It is a situation in which we can apply Markov’s inequality.*0318

*Let me go ahead and write down Markov’s inequality and we will see how to apply it to the situation.*0323

*Markov’s inequality, remember, we said the probability that Y is greater than*0328

*or equal to a certain value A is less than or equal to the expected value of A, the mean of Y divided by A.*0334

*In this case, the first question here we want to estimate the chance that a student is carrying more than $100.*0343

*We want to find the probability that a student will have more than 100 and I’m going to fill in the expected value of Y,*0350

*that is the average value of Y which we have been given as $20.00.*0358

*It is 20 for the expected value and then the value of A that we are using is 100, 20/100 simplifies down to 1/5.*0362

*The answer to our first question there is the probability is less than or equal to 1/5.*0372

*Notice that I'm being very careful here not to say that it is equal to 1/5.*0379

*It might be considerably less than 1/5.*0384

*I do not know if it is less than or less or equal to.*0387

*To be safe, I’m just going to say the probability is less than or equal 1/5.*0390

*That is really all we can tell using Markov’s inequality.*0394

*We also want to estimate the chance that a student is carrying less than $80.00.*0399

*That is the other direction of Markov’s inequality, the probability that Y is less than the value A.*0405

*Reversing Markov’s inequality, that probability is greater than or equal to 1 – E of Y, the expected value /A.*0414

*That is the equation that we learned back there on the first side.*0423

*In this case, our A is 80.*0426

*The probability that Y is less than 80 is greater than or equal to 1 -, the expected value is still 20, that was given in the problem.*0430

*20/80 is ¼, this is equal to 1 – 1/4 which is ¾.*0440

*The probability is greater than or equal to ¾.*0455

*What we can say there, if we meet a random student is that the probability that she will have less than $80.00 is at least ¾.*0462

*At least 75% chance she has less than $80.00 in cash on her.*0471

*That answers both of our questions here.*0477

*Note that I could not give you exact probabilities.*0480

*In either case, I have to give you just inequality because that is all Markov’s inequality gave you.*0482

*Let me remind you how we did that.*0489

*I start off with the basic formula of Markov’s inequality.*0490

*This is just the same equation we got on the first slide here.*0493

*Since, we are asked about carrying more than $100, I filled in A =100 here.*0497

*A equals 100 here and the expected value that is the average of $20.00.*0502

*I plugged that in right here and then I just simplify that down 20/100 is 1/5.*0509

*It is important to get the inequality the right way.*0514

*What we can say here is that it is unlikely that a student will have more than $100 and*0518

*how unlikely it is less than 1/5 or less than 20% chance.*0524

*On the other side, we are asked about the chance that she is carrying on less than $80.00.*0529

*I’m using the less than form of Markov’s inequality, that was the second version that I gave you.*0536

*We were told that A equals 80, we plug in the expected value is 20.*0543

*Simplify that down to ¾ and what I can tell you is that if I meet a student,*0548

*there is at least a 75% chance that she will have less than $80.00.*0554

*Let us keep that going with the next example here.*0561

*Here we have a rental car agency, they are doing some statistical analysis of their cars.*0566

*They require that customer return cars after a weeks rental, they put an average of 210 miles on the cars.*0574

*We just had a new customer check out a car for a week and we want to estimate the probability*0582

*that the customer will put more than 350 miles on the car.*0589

*This is a classic Markov’s inequality problem, let me write down Markov’s inequality to get us started.*0594

*The probability that Y is greater than or equal to A is less than or equal to the expected value or the mean value of Y/A.*0600

*In this case, we want to estimate the probability that he will put more than 350 miles.*0610

*The 350 is the A there, 350 is less or equal to.*0617

*The expected value is the average number of miles that these customers are putting on the cars, that is 210/350.*0623

*If I divide top and bottom by 70 there, we also have a factor of 70, that would simplify down to 3/5.*0634

*That is as simple as it is going to get.*0644

*The probability is less than or equal to, 3/5 is 60%.*0648

*I will write that as a percentage.*0653

*What that tells me, what we can tell our associates for this rental car company is that this particular customer,*0657

*there is less than 60% chance that this customer is going to put 350 miles on the car.*0664

*That is the best we can say, we can never give an exact answer with Markov’s inequality.*0675

*We can just put a bound on it, above and below.*0680

*Here, we put an upper bound of 60% chance that the customer is going to put that many miles on the car.*0682

*To show you how I got that, start out with the basic version of the Markov’s inequality.*0689

*I figured out that the A I was looking for was 350, that came from the stem of the problem here.*0694

*I plugged that in, 350 in both places.*0701

*210 is the average number of miles the customers put on the car, that is the expected value of the random variable.*0704

*That simplifies down to 3/5 and the important thing here is that you give your answer as an inequality.*0713

*You do not want to just say 60%.*0719

*When I taught probability, a lot of times my students will just try to give me a number as an answer.*0721

*This 60% as the answer and it does not tell me what I want to know because you are saying that is equal to 60%.*0727

*And we do not know that, all we know is that it is less than or equal to 60%.*0736

*That is all Markov’s inequality tells us.*0740

*In example 3 here, we have done some tracking of the history of earthquakes in California.*0745

*Apparently, there is a major earthquake in California on average, once every 10 years.*0752

*We want to describe the probability that there would be an earthquake in the next 30 years.*0757

*Maybe, we are planning a major investment in California and we are wondering*0763

*how likely it will be that there will be an earthquake in the next 30 years.*0766

*Let me described carefully here what the random variable is,*0774

*because I think it is a little less obvious in this one than in some of the previous ones.*0778

*Y here is going to be the waiting time until the next major earthquake.*0782

*What they have really told us when we say that it occurs on average once every 10 years,*0802

*they told us that the average waiting time for one earthquake to the next is 10.*0807

*E of Y is equal to 10, that is what they have given us.*0813

*We want to find the probability that there will be an earthquake in the next 30 years.*0817

*The probability that our waiting time for the next earthquake is less than 30, that is what we are trying to calculate.*0823

*Within the next 30 years is what we are trying to find.*0836

*We are going to use Markov’s inequality but since we are trying to estimate the probability that Y is less than a cut off,*0842

*we are going to use the version of Markov’s inequality, the reversed version of Markov’s inequality,*0850

*P of Y less than A is greater than or equal to 1 – P of Y/A.*0858

*In this case, our A is 30.*0866

*This is 1 – E of Y which is 10/A is 30.*0869

*I put that that was equal to, I have committed the crime in Markov’s inequality*0877

*that I have been telling you not to do which I said the probability was equal to something.*0882

*We never know that for sure, not for Markov’s inequality.*0887

*We always get a one sided bound so the probability is greater than or equal to 1 – 10/30*0890

*and that simplifies down to 1 -1/3 which is 2/3.*0900

*What we can say here is that the probability that Y is less than 30, remember,*0906

*that is the probability that we will have an earthquake in the next 30 years is greater than or equal to 2/3.*0914

*That is the conclusion we can make from the information we are given and from Markov’s inequality.*0921

*What that saying is that it is pretty likely that there will be an earthquake in California sometime in the next 30 years, or at least a 2/3,*0928

*or 67% chance that there will be an earthquake in California in the next 30 years.*0938

*To show you how I figure that out, the important thing here was setting up the random variable.*0945

*We said Y is going to be the waiting time, how long the wait until we see the next major earthquake.*0949

*Since, they occur once every 10 years on average, that does not mean they occur with clockwork regularity every 10 years.*0956

*It just means they occur on average, once every 10 years.*0964

*The expected value of that variable is 10.*0966

*We want to find the probability that it is less than 30 because if it is less than 30*0971

*that means we will have an earthquake in the next 30 years sometime.*0976

*According to reversed formula for Markov’s inequality, that is a bigger than 1 - the expected value/30.*0980

*Remember, I use equals and that was a mistake, it is really greater than or equal to, that simplifies down to 2/3.*0990

*Our final conclusion here is that the probability is greater than 2/3, greater than or equal to 2/3.*0998

*I think there is at least a 67% chance that we will have an earthquake in the next 30 years here in California.*1006

*For example 4, we have a factory that produces batches of 1000 laptops.*1015

*I guess each day they runoff a batch of 1000 laptops and send them out for distribution.*1020

*They find that on average they will do some testing, on average 2 laptops per batch are defective.*1026

*They have some kind of a serious defect in them.*1033

*We want to estimate the probability that in the next batch, fewer than 5 laptops will be defected.*1036

*Again, this is a Markov’s inequality problem.*1044

*Let me go ahead and set up the generic inequality for Markov.*1047

*That is the probability that Y is greater than or equal to A is less than or equal to the expected value of Y divided by A.*1054

*In this case, we want to reverse that because we want estimate the probability that fewer than 5 laptops will be defective.*1064

*Let me go ahead and do the reverse version of that.*1075

*The probability that y is less than A is greater than or equal to 1 – E of Y/A.*1078

*We are just taking Markov’s inequality and then taking the complement of it.*1087

*That should not be something you really have to memorize, it should be something you can figure out from the original Markov’s inequality.*1090

*In this case our A is 5, we want the probability that fewer than 5 laptops will be defective.*1097

*It is greater than or equal to 1 -, E of Y is the expected value or the mean or the average number of laptops per batch.*1104

*We are given that that is 2, this is 1 - 2/5 here.*1114

*I'm filling in 5 for the value of A because that is what we had on the left hand side.*1121

*1 - 2/5 is 3/5 and we could simplify it, we can convert that into a percentage.*1125

*The probability here is greater than or equal to 3/5 is 60%.*1134

*If you are a company manager, and you got some quality control specifications*1141

*that says you cannot have any more than 5 laptops per batch be defective,*1148

*which you can say is that in the next batch, there is at least a 60% chance that we would not have 5 or more laptops defective.*1154

*That is the best you can say with Markov’s inequality.*1164

*You cannot put a precise value on the probability, you can just give a lower bound and say*1167

*at least 60% of the time that we will have more than 5 laptops be defective.*1172

*Where that came from was, I started with the original version of Markov’s inequality and then*1179

*I realize that I needed to turn this around because the original version has Y being bigger than the cutoff A.*1186

*In this case, I want to estimate the probability that fewer than 5, that is why less than A.*1194

*That is the reverse of Markov’s inequality.*1201

*The probability that Y is less than A is greater than or equal to 1 – E of Y/A.*1204

*Plug in A equals 5 because that is coming from the stem of the problem.*1210

*I plugged in the expected value that is the average number of laptops per batch, where that come from, that comes from here.*1215

*That is where that 2 come from.*1223

*Let me simplify the numbers down to 3/5 which is 60%.*1225

*What I can say from this is that the probability is at least 60% that fewer than 5 laptops are defective in the next batch.*1229

*If you are the factory manager, you can decide whether that is acceptable.*1250

*Are you willing to accept the 60% chance of having fewer than 5 defectives*1254

*or do you need to tighten your quality control procedures based on that probability?*1258

*Let us keep moving onto the next example here.*1265

*In our final example, we got a grocery store that selling an average of 30 cans of tuna per day.*1269

*We want to estimate the probability that it will sell more than 80 cans tomorrow.*1275

*You are the manager of this store and you are worried about whether you are going to run out of your stock of cans of tuna tomorrow*1278

*Should you order some more or can you hold out for a couple more days?*1286

*Again, this is kind of a classic Markov’s inequality problem.*1291

*Something that makes it Markov’s inequality, I did not mention this on some of the previous examples,*1294

*is that we are only calculating values that are going to be positive here.*1300

*The number of cans of tuna that a grocery store is going to sell in any given day, that is going to be positive.*1306

*It could be 0, it could be significantly higher than 0 but it is almost certainly not going to be negative.*1312

*We are not going to be having many cans of tuna return.*1318

*This is a positive quantity, it is okay to use Markov’s inequality.*1323

*Our Y here is the number of cans of tuna sold each day.*1336

*Let us write down our Markov’s inequality.*1348

*The probability that Y is less than or equal to A is greater than or equal to E of Y divided by A.*1351

*That is just our generic formula for Markov’s inequality, we learned that back in the first slide of this lecture.*1360

*In this case, our A is our cutoff value.*1365

*The probability in this case, we are trying to estimate the probability that it will sell more than 80 cans.*1369

*I wrote down my Markov’s inequality, I wrote the inequality so I got them switched here.*1380

*Probably, the Y is greater than or equal to A is less than or equal to E of Y/A, that is the original version of Markov’s inequality.*1386

*The one that I was kind of channeling there was the opposite one,*1394

*the probability that Y is less than A is greater than or equal to 1 - E of Y/A.*1398

*We have to look at our problem and figure out which one of those is going be relevant.*1407

*In this case, we want the probability that it will sell more than 80 cans tomorrow so that is a greater than or equal to.*1410

*The probability that Y is greater than or equal to 80 is less than or equal to*1418

*the expected value not the average number of cans it sells on a normal day, that is 30 cans.*1425

*Let me fill in 80 here for my value of A.*1433

*If I simplify 30/80 that this reduces to 3/8.*1439

*If you convert that into a percentage, that is very easy to convert into a percentage.*1443

*It is halfway between 1/4 and ½.*1448

*It is halfway between 25% and 50% , that is a 37.5%.*1452

*The probability is less than or equal to 37.5%.*1461

*If you are a store manager and you are wondering, maybe, you got 80 cans of tuna in stock.*1471

*You are worry about whether you are going to sell out tomorrow, maybe you are going to need to order some more.*1476

*The probability that you are going to sell all 80 of those cans is at most 37.5%.*1481

*You do not know exactly what the probability is, because Markov’s inequality never tells you*1488

*an exact value but it tells you that it is less than or equal to 37.5%.*1492

*Let me recap how we did that.*1499

*We started with the basic version of Markov’s inequality, P of Y is greater than or equal to A less than E of Y/A.*1501

*I went ahead and wrote down the reverse version, because when I wrote down the basic version,*1509

*I accidentally switched the inequalities.*1513

*I wrote down the reverse version just to make sure that we are keeping everything straight there.*1515

*Since we are selling more than 80 cans that means we want the positive version of Markov’s inequality, the more than version.*1522

*I fill that in with A equals 80, filled in A = 80 in the denominator here.*1532

*The E of Y that is the average number of cans sold, that is the 30 from the problem stem.*1537

*That simplifies down to 3/8 or 37.5%.*1545

*What Markov’s inequality tells us is that the probability is less than or equal to 37.5%.*1548

*That less than or equal to is really an important part of your answer.*1556

*You are giving an upper bound, you are not saying it is equal to 37.5%.*1561

*You are just saying that is the most it could possibly be.*1565

*That wraps up our lecture on Markov’s inequality.*1569

*Next, we are going to be talking about Tchebysheff's inequality which is a little bit stronger than Markov’s inequality.*1572

*You usually get a stronger version with Tchebysheff's than Markov*1578

*but after a little bit work form to Tchebysheff's inequality, you have to know the standard deviation.*1582

*Markov’s inequality, we just had to know the expected value or the mean of the random variable.*1588

*I hope you stick around and we will learn about Tchebysheff's inequality in the next video.*1594

*In the meantime, you have been watching the probability videos here on www.educator.com.*1597

*My name is Will Murray, thank you for watching today, bye.*1603

1 answer

Last reply by: Dr. William Murray

Fri Nov 10, 2017 12:48 PM

Post by Natalia Stein on November 7 at 10:04:44 AM

What happens when the average of 5 laptops are usually defective in each batch?