For more information, please see full course syllabus of Probability

For more information, please see full course syllabus of Probability

### Random Variables & Probability 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
- Intuition
- Intuition, Cont.
- Definition
- Probability Distributions
- Example I: Probability Distribution for the Random Variable
- Example II: Probability Distribution for the Random Variable
- Example III: Probability Distribution for the Random Variable
- Example IV: Probability Distribution for the Random Variable
- Example V: Probability Distribution for the Random Variable

- Intro 0:00
- Intuition 0:15
- Intuition for Random Variable
- Example: Random Variable
- Intuition, Cont. 2:52
- Example: Random Variable as Payoff
- Definition 5:11
- Definition of a Random Variable
- Example: Random Variable in Baseball
- Probability Distributions 7:18
- Probability Distributions
- Example I: Probability Distribution for the Random Variable 9:29
- Example II: Probability Distribution for the Random Variable 14:52
- Example III: Probability Distribution for the Random Variable 21:52
- Example IV: Probability Distribution for the Random Variable 27:25
- Example V: Probability Distribution for the Random Variable 34:12

### Introduction to Probability Online Course

### Transcription: Random Variables & Probability Distribution

*Hi and welcome back to the probability lectures here on www.educator.com.*0000

*My name is Will Murray, and today, we are going to talk about random variables.*0004

*The idea that goes along with that is the probability distribution.*0008

*We are going to learn what those terms mean, let us jump right into it.*0012

*I'm going to start with the intuition for random variable*0017

*because a formal definition of random variable is not really that illuminating.*0021

*We will start out with the intuition, try to give you an idea of roughly what they mean,*0026

*then I will give you the formal definition.*0030

*The intuition for a random variable is it is a quantity you keep track of during an experiment.*0032

*We are going to use Y for random variable.*0040

*I want to start out with an example right away.*0045

*Let us say we are going to play the World Series and the Yankees are going to play the Giants for 7 games.*0048

*That is not quite how the actual world series run.*0054

*Before I get a bunch of angry comments from a bunch of sports fans,*0056

*I know that some× in the World Series, one team wins all four games right away.*0060

*It does not actually run for 7 games.*0068

*We are not playing under those rules.*0070

*We are going to play 7 games, all 7 games no matter what, no matter who wins the first 2 games.*0072

*What can happen there is, there are7 different games, *0078

*each one can have 2 possible outcomes because the Yankees can win and the Giants can win.*0082

*There are really 2⁷ possible outcomes.*0086

*There are 128 possible sequences of events in the World Series.*0089

*Honestly, what we have been really only care about is the number of games that one team wins vs.*0095

*the number of games that the other team wins.*0101

*We could say, for example, that Y is going to be the number of games that the Yankees win.*0104

*What we are really care about is whether Y is more or less than 4 *0111

*because whichever team wins 4 more games wins the world series.*0114

*We certainly do not really care who wins the first game vs. the second game,*0118

*what we care about is the total number of games that each team wins.*0123

*For example, one possible way the World Series can go would be,*0128

*if you are keeping track of the Yankees, you can have win-win, lose-lose, win-lose-win.*0133

*That is the possible outcome of the World Series.*0143

*What we would keep track of their though is that there are 4 wins.*0146

*We would say Y of that outcome would be 4.*0152

*That is really all that matters, it does not matter the order in which the wins occur.*0155

*That is the intuition for random variable is you are keeping track of a certain number during an experiment.*0160

*At the end, your outcome gives you a certain value.*0167

*Let us do another example to try to understand that intuition a little more before I give you the formal definition.*0171

*Another way to think about random variables is to think about a payoff on an experiment.*0179

*Meaning depending on what happens, you get paid a certain amount of money.*0185

* This is often useful if you are thinking about gambling games in a casino.*0190

*Depending on how the cards come out, the dice come out, whatever, you get paid a certain amount.*0194

*For example, here is a very simple gambling game.*0199

*You are going to draw a card from a 52 card deck, if you drawn an ace then I will pay you $1.00.*0203

*If you draw a 2 then I will pay you $2.00, all the way up to I you draw a 9 then I will pay you $9.00.*0209

*If it is a 10, you have to pay me $10.00.*0217

*If it is a jack, queen, king, those are called face cards then you have to pay me $10.00 for any of those cards as well.*0221

*The quantity we can keep track of here is the amount of money that you stand to make on this experiment.*0232

*Notice here that this Y could be positive or negative.*0238

*For example, Y if you draw an ace then you stand to make a $1.00 off this experiment, that is a positive outcome for you.*0242

*Y if you draw a 2, would be $2.00, all the way up to Y of 9, if you draw a 9, you get $9.00.*0252

*But if you draw 10 then you have to pay me $10.00.*0264

*From your perspective, that is -$10.00 for you.*0269

*Y of 10 would be -10 and Y of a jack would be -10, and so on.*0273

*Any face card, you have to pay me $10.00.*0282

*We think of the payoff and from your perspective that is a negative payoff for you.*0285

*The way you want to think about this random variable is it is the amount of money that you make from this experiment *0291

*which could be positive or negative, depending on whether you go away richer or whether you have to pay me and you walk away poor.*0299

*Those are some rough, intuitive ideas to keep in mind as we get to the formal definition of random variable,*0306

*which is that it is a function from the sample space to R.*0314

*R is the set of real numbers here.*0319

*Remember, the sample space is the set of outcomes for an experiment.*0322

*The random variable number, if you want to think about it as a payoff then for each outcome there is some kind of payoff.*0340

*There is a real numbers worth a payoff.*0346

*It is a function that takes in an outcome and it gives you back a number which you can think of is the amount you get paid.*0349

*Or it is the quantity that you are keeping track of during the experiment.*0359

*To return to our first example there which was the World Series, how many games do the Yankees win?*0364

*I have listed some different possible outcomes of the World Series.*0370

*We are assuming that we are going to play all 7 games in the World Series,*0374

*even if somebody has already wrapped up the best of 7 early.*0378

*We are going to play all 7 games, no matter what.*0383

*If we are kind of looking at it from the perspective of the Yankees, if it is win-win-win, lose-lose-win-lose, there are 4 wins there.*0385

*The Y of that outcome is 4.*0396

*If we lose the first 6 games and we win the last game, the Y of that outcome is 1.*0399

*If we alternate winning and losing, win-lose, win-lose, win-lose-win, that is also 4.*0406

*In the sense of comparing that to the first outcome there, they are the same as far as Y is concerned *0412

*because the Y is going to be the same number either way.*0419

*Remember, there is 2⁷ possible ways that the World Series could go.*0423

*I’m not going to write them all down.*0427

*Each one of them has a number associated to it, from 0 to 7 depending on how many games the Yankees win there.*0429

*I want to move on to my next definition which is the probability distribution.*0439

*The notation gets a little confusing here because there is a lowercase letters and capital letters.*0443

*Here you want to think of y here as a number, that is a possible value of the random variable.*0449

*The Y here is the actual random variable.*0458

*This is what gets a little confusing because we will say Y=y.*0468

*What we are really asking there is when is the random variable going to take on that particular number or that particular value?*0472

*You think of the y as being the value and Y is the actual variable.*0482

*What is the probability that the random variable has a particular value?*0488

*This P is the probability that the random variable takes on a particular number or takes on a particular value.*0494

*We try to find that probability and we add up all those probabilities over all the outcomes that lead to that value.*0505

*We say that that is the probability of that value.*0513

*This problem will make more sense after we do some examples.*0516

*Stick around for examples, if it is a little confusing to you right now.*0519

*What we are doing here, what this notation means is now the sample space is the set of all the outcomes.*0523

*We look at all the outcomes in the sample space for which the random variable has that particular value *0531

*and then we add up the probabilities of all those outcomes.*0538

*That is what this formula means.*0542

*It will make more sense after we do some examples.*0543

*This function, we think of this as being a function P of y.*0548

*It is the probability distribution of the random variable Y.*0552

*We will talk about calculating P of different numbers.*0556

*I think that will make some sense after you see some of these examples.*0560

*Let us jump in and do some examples in all of these notations to make a little more sense.*0566

*In our first example, we are going back to the one that I mentioned earlier in the lecture*0571

*where you draw a card from a standard 52 card deck.*0576

*If it is an ace -9, I will pay you that amount.*0580

*If you draw an ace, I’m going to pay you $1.00.*0583

*If you draw a 2, I will pay $2.00.*0585

*All the way up to, if you draw a 9 I will pay $9.00.*0587

*If it is a 10 then all of a sudden the tables are turned and you have to pay me $10.00.*0591

*From your perspective, that is -$10.00.*0597

*If it is a jack, queen, king, those are called face cards, then you still have to pay me $10.00.*0599

*The question here is what is the probability distribution for this random variable?*0605

*What I'm really asking here, what this kind of question is asking is *0612

*what is the probability of getting different possible values for the random variable?*0616

*What is the probability that you will collect exactly $0.00 from this experiment?*0624

*The probability that this random variable will take the value 0.*0631

*In this case, there is no way that it is going to be exactly 0 because if you get a certain set of cards, *0634

*you are going to have to pay me.*0641

*If we get a different set of cards, I'm going to have to pay you.*0643

*There is no way that this experiment can wind up being worth $0.00 from you.*0647

*You are going to win something, you are going to lose something, you are not going to break even on this experiment.*0652

*Let us look at the probability of 1.*0657

*P of 1 is the probability that our random variable takes the value 1.*0661

*Let us think about all the ways that you can make exactly $1.00 on this, that means you have to draw an ace.*0669

*There are 4 aces in a 52 card deck and the probability that you are going to make exactly $1.00 on this is exactly 1/13.*0676

*The probability that you make $2.00, that is the probability that the random variable takes the value 2.*0689

*There are four 2s in the deck, you have a 4/52 chance which simplifies again down to 1/13.*0698

*It is similar to that all the way up through 3, 4, 5, 6, up to 9, the probability of 9, again it is the probability that Y is equal to 9.*0708

*There are four 9 in a 52 cards and that is 1/13.*0720

*It suddenly changes, there is no way you can get 10 because if you draw a 10 from the deck then you have to pay me.*0730

*That does not count as getting $10.00 for you.*0739

*That would be the probability that Y is equal to 10 and there is no way that you can get $10.00 out of this game, that is 0.*0743

*The probability, there is one more possible value that this game can take for you know which is -10.*0752

*That happens when you draw a 10 or you draw a face card, any of those, the result in you having to pay me $10.00.*0759

*How many different cards are there?*0773

*There are four 10, jacks, queens, kings, there are 16 cards here that will give you a net loss of $10.00*0775

*That simplifies down to 4/13.*0788

*That is your probability of getting -$10.00 out of this game, if you are losing $10.00 as you play this game.*0793

*That is our full probability distribution, I will put a box around the whole thing here because really that whole thing is our answer.*0801

*The probability distribution means you are thinking of all the different possible values of y and*0811

*you are calculating the probability of Y, of each one of those values.*0818

*And that is what we did here.*0822

*To recap that, the possible values of y, I threw in 0.*0824

*It turned out that there is no way that you can make exactly $0.00 out of this game.*0828

*The probability of getting 0 is 0.*0833

*Then, I calculated the probability of 1 through 9.*0837

*For each one of those, there were 4 cards that can give you that particular payoff and *0840

*that is 4/52 giving us 1/13 for each one of those.*0847

*There are really 9 different possibilities there.*0855

*They all have probability 1/13.*0858

*And then, I went ahead and included the probability of 10 although there is no way that you can get $10.00 out of this game.*0860

*That is why we got 0 there.*0868

*I also had to include -10 because there is a possibility that you might lose $10.00 off this game.*0870

*There are 16 cards in the deck that will give you a loss of $10.00.*0877

*When I simplify 16/52, I get 4 out of 13.*0881

*That is the probability distribution for that random variable on that experiment.*0886

*In our second example here, we are going to flip a fair coin 10 ×.*0894

*Y is going to be the number of heads and I want to find the probability distribution for this random variable.*0897

*We are going to calculate P of y, where y is all the possible values that Y could be.*0906

*It is the probability that Y could be equal to that particular value y.*0915

*Let me note before we start, the possible values that y could be, that is the number of heads that we can see here.*0923

*And the fewest heads, we could possibly get be 0.*0931

*The most heads we can possibly get would be 10, if all 10 flips come out to be heads.*0934

*Let us think of on the probability of any particular value.*0939

*Let me think about that, to get exactly y heads we must,*0948

*Here is how you can think about that, we must fill in 10 blanks 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 with tails and heads.*0959

*We know how many of the each because we would have to fill in y heads, yh.*0984

*That would mean that the remaining 10 - y blanks, 10 – yt would all have to be tails.*0993

*Let us think about that, we are choosing y of those 10 blanks to be heads.*1002

*We must choose y blanks to be heads.*1012

*Once we made that choice, there is nothing more to decide because automatically then, all the remaining blanks would have to be tails.*1022

*Let us think about how many ways there are to do that.*1033

*There are 10 choose y ways to do that.*1039

*This is something we learn earlier on in the probability lecture series here on www.educator.com.*1050

*We had a chapter on making choices.*1056

*Probably, that chapter was on ordered vs. unordered, with replacement vs. without replacement.*1059

*This is really an unordered, without replacement selection.*1066

*The formula that you use for that kind of choice is 10 choose y.*1075

*You can look that up on the earlier lectures, if you do not remember how to do that.*1081

*But there is 10 choose y ways to do that.*1086

*Suppose you have a particular way of doing that, each particular way of filling in the blanks,*1089

*let us say for example 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.*1102

*Let us say, for example y =3.*1108

*You choose 3 places, head-head and head.*1111

*You choose 3 places that mean all the others have to be tails.*1115

*What is the probability of that exact arrangement of heads and tails coming up in 10 flips?*1119

*It means you got to get a tail the first flip then a tail, then a head, then tail-tail, head-head, tail-tail-tail.*1127

*Each one of those has ½ chance of occurring.*1133

*To get that entire sequence, that exact sequence, each one has a ½ ⁺10 or 1/2 ⁺10 chance.*1136

*Any particular exact sequence has a 1/2 ⁺10 chance of occurring.*1151

*The probability of getting all those sequences is 10 choose y × 1/2 ⁺10.*1161

*10 choose y, these are combinations, that is the notation for combinations that I'm using there.*1173

*You could also use the notation C of 10 choose y, if you prefer that notation.*1180

*We use that also in the earlier lecture here on www.educator.com.*1188

*10 choose y divided by 2 ⁺10, that is the total probability that we are going to get exactly y heads.*1193

*Let me remind you the range here.*1203

*This is for all possible values of y from 0 up to 10.*1205

*That is our probability distribution, that is our probability of getting any particular value of y, as y ranges from 0 to 10.*1211

*Let me go back over that and make sure that everything is still clear.*1224

*We are trying to find the probability of any particular value of y, which means *1228

*the probability that our random variable will take the value exactly y, *1232

*which means it will get exactly y heads when we flip the coin 10 ×.*1238

*When we think about that, it means that you are filling in 10 blanks with y heads and 10 - y tails.*1244

*If you think about the number of ways to do that, there are 10 choose y ways to fill in exactly y heads.*1253

*Once you fill in the heads then you have to fill in all the remaining places with tails.*1263

*That was an unordered choice, that was without replacement.*1270

*If you do not remember what those words mean, there is an earlier video here on the educator series on probability.*1273

*You can just go back and look those up and you will see that this is what we are talking about.*1280

*Each one of those ways, each one of those exact sequences has a 1/ 2 ⁺10 *1283

*probability of coming up when you flip a coin 10 ×.*1290

*The total probability is 10 choose y divided by 2 ⁺10 or 10 choose y × 1/ 2 ⁺10.*1294

*That is our probability distribution, that is our probability of getting exactly y heads for any number y between 0 and 10.*1302

*In our third example here, we are going to roll a dice repeatedly until we get a 6 showing.*1314

*We want to let y be the number of rolls that it takes for us to see our first 6.*1320

*We want to find the probability distribution for this random variable.*1326

*Let us go ahead and calculate.*1330

*First of all, the range that we can get then different values.*1332

*It could be that we get very lucky that we get a 6 right away.*1337

*Let me write this in that form because our y could be 1, if we get 6 right away.*1342

*If we miss the 6 in the first roll, we get it on the second roll, then it could be 2, it could be 3.*1350

*Actually, this could go on indefinitely because we really do not know how many rolls it might take.*1357

*If we are very unlucky, it could take as 150 rolls before we see the first 6.*1363

*Another way to say this is, 1 is less than or equal to y less than infinity.*1369

*Y could be potentially any positive integer here.*1375

*We are going to have to investigate the probability of each one of those values.*1379

*Let us think about what those probabilities are.*1384

*The probability of getting 1, let me write that in P notation.*1387

*The probability that the random variable is equal to 1.*1392

*That means you get a 6 on your very first roll.*1396

*There is a 1/6 chance that you get a 6 on your very first roll.*1400

*What about the probability that you get 2?*1405

*That is the probability that Y is equal to 2.*1409

*That means you get a 6 on the second roll.*1413

*Think about that, that means you must not have gotten the 6 on the first roll*1416

*because if you got a 6 on the first roll, you would have stop.*1421

*In order to get a 6 exactly on the second roll then you must get a 6 on the first roll, *1424

*there is a 5/6 chance that you are going to fail on that first roll.*1432

*And then, you must get a 6 on the second roll which means there is a 1/6 chance of getting that.*1436

*The probability of taking exactly 2 rolls here is 5/6 × 1/6. *1444

*Or about the probability of getting exactly 3 rolls.*1452

*The probability that our random variable takes the value 3.*1456

*To get 3 rolls that mean you missed getting a 6 on the first 2 rolls.*1462

*It is 5/6 to miss it on the first roll.*1465

*It is 5/6 again, to miss it on the second roll because if we get it on the first and second roll,*1469

*you are not going to get on to 3 rolls, × you have to get it in the 3rd roll so there is 1/6 there.*1475

*This continues here, the probability of taking exactly y rolls which is the probability that Y is equal to the number of y.*1484

*You are going to multiply together a bunch of 5/6 and how many you are going to multiply that, represents losing not getting a 6 on the first y -1 rolls *1497

*because you want to get a 6 exactly on the last roll, on the yth roll.*1509

*It is 5/6 ⁺y -1 × 1/6.*1515

*This represents failing on the first y -1 rolls and this represents succeeding on the yth roll.*1523

*That is what it takes, in order to get exactly y rolls.*1545

*That is our generic formula and let me go ahead and remind you that *1551

*the range for y was any number bigger than or equal to 1 and less than infinity.*1556

*It could be arbitrarily big, it could take thousands and thousands of rolls, if you are very unlucky to get our first 6 here.*1565

*Let me remind you of the steps there.*1576

*We want the probability that is going to take exactly y rolls to get a 6.*1578

*For example, to get exactly one roll, to get a 6, there is a 1/6 chance that we are going to get a 6 on the very first roll.*1585

*2 rolls, we have to get one of the other five numbers for the first roll and then get a 6 for the last roll, for the second roll.*1593

*Three rolls, twice in a row we have to get one of the other five numbers.*1602

*On the third roll, we have to get our 6.*1607

*In general, to get exactly y rolls, we would have to get one of the other five numbers, *1610

*in other words fail to get a 6 on the first y -1 rolls.*1617

*On the very last roll, the yth roll, we have to get a 6, there is a 1/6 chance.*1622

*We put those together as 5/6 ⁺y-1 × 1/6.*1628

*Our range there is from 1 up to potentially as large positive integer as you can imagine there.*1634

*In example 4, we are going to keep track of a soccer match here, *1647

*actually 3 soccer matches between Manchester United and Liverpool football club.*1650

*In any given match, it tells us Liverpool is the stronger team this season.*1657

*They are twice as likely to win as Manchester, I hope I’m not upsetting a huge range of football fans here.*1662

*There are no ties, I’m eliminating ties here.*1668

*We are going to play penalty kicks or something, until we have a winner of every match.*1672

*Let y be the number of matches that Liverpool wins.*1677

*We are going to look at things from Liverpool’s perspective.*1680

*We want to know what is the probability distribution for this random variable?*1683

*The first thing to notice here is that, since Liverpool is twice as likely to win as Manchester, *1689

*in any given match, the probability in each particular match, Liverpool wins with probability,*1695

*If Liverpool is twice as likely to win, they must have a 2/3 chance of winning*1716

*because that would give Manchester 1/3 chance of winning.*1722

*The 2/3 is twice as likely as 1/3.*1725

*Manchester wins with probability is 1/3.*1730

*That is for any particular match, let us try to figure out the probabilities that Liverpool will win a certain number of matches.*1735

*If they are playing three matches then Liverpool might lose all 3, it might win 1, it might win 2, it might win all 3.*1742

*We are going to have to calculate the value 0, 1, 2, and 3.*1751

*The probability of 0, P of 0, that is the probability that Liverpool wins exactly 0 matches.*1755

*In order to win 0 matches, they are going to have to lose all three matches.*1765

*Any given match, they lose a probability 1/3.*1771

*The odds of them losing 3 in a row is 1/3³ which is 1/27.*1773

*That is the probability that Liverpool walks away from three matches without a single win.*1781

*The probability of 1, that is the probability that their total winnings are one match.*1787

*Let us think about that, how can Liverpool win one match?*1796

*One way to do it would be if they win the first one and then lose twice.*1799

*They could lose-win-lose, they could lose-lose and win the final match.*1804

*There are three different possibilities there but each one of those has Liverpool winning one match and losing 2.*1813

*In each one of those three possibilities, there is a 1/3 chance that they will win their relevant match.*1821

*There is a 1/3² chance that they will lose the 2 relevant matches.*1832

*A 2/3 chance that they will win the match that they are supposed to win.*1837

*If we multiply those all together, the 3 and 2/3 give us just 2 × 1/3 × 1/3 that is 2/9.*1844

*2/9 is the total probability that Liverpool will win exactly one match there.*1858

*How about the probability that they will win exactly two matches?*1863

*What is the probability that y is equal 2?*1869

*How can we win two matches?*1872

*They could win the first 2 and then lose, they could win-lose-win, they could lose and then come back and win 2 in a row.*1875

*There are 3 ways that can happen.*1886

*Any particular one of those ways, the likelihood that I have to win 2 particular matches,*1889

*there is a 2/3 chance for each one of those.*1895

*We have to lose a particular match, there is 1/3 chance that they will lose whichever match they are supposed to lose.*1898

*If we multiply those together, the 3 and 1/3 cancel each other out.*1905

*We get 4/9 there, that is the probability that Liverpool will win exactly 2 matches out of their 3 match series there.*1910

*Finally, what is the probability that Liverpool is going to win all three of them?*1921

*Probability that y=3?*1927

*To do that, Liverpool would just have to win-win-win.*1932

*The only way to do that is to win three matches in a row.*1937

*Each one, they have a 2/3 chance, 2/3³ is 8/27.*1940

*8/27, we have our whole probability distribution here.*1952

*We have got the 4 values, y goes from 0 to 3, the four numbers of matches that Liverpool might win.*1961

*We have our probabilities for each one.*1968

*Let me remind you how we figure that out.*1972

*First of all, we know that Liverpool is twice as likely to win as Manchester, *1975

*which means that Liverpool is going to win with probability 2/3, Manchester with probability 1/3.*1980

*We said there are no ties here, we are not allowing ties, that would just make it too complicated.*1986

*We figure out the probability that Liverpool is going to win 0 matches that means they have to lose 3 × in a row.*1992

*There is a 1/27 chance of that.*1998

*How can Liverpool win exactly one match?*2001

*There is three different ways they can do that.*2003

*For each one of those ways, the probability is 1/3 × 1/3 × 2/3, and some of ordering of that.*2005

*You multiply those together, that is where the 2/9 comes from.*2015

*How can we win exactly 2 matches?*2019

*There is three ways they can win exactly 2 matches.*2021

*For any one of those configurations, the probability is 2/3 × 2/3 × 1/3.*2024

*That is where we get the 4/9. *2031

*Finally, the probability that they will win all three matches is they would have to win all three in a row.*2033

*There is a 2/3 × 2/3 × 2/3 chance that they will win all 3.*2041

*That gives us 8/27 for the last value in our probability distribution their.*2046

*In the last example here, you are going to play a game with a friend.*2054

*You are going to do a little gambling with your friend.*2057

*You and your friend are each going to flip a coin, you flip your coin and your friend flips his coin.*2059

*If both the flips come out heads, then your friend has to pay your $10.00.*2065

*If they are both tails then he pays you $5.00.*2069

*If the coins match each other then your friend is going to pay you.*2073

*That is a good thing for you.*2077

*If the coins do not match each other, meaning you get a tail and he gets a head, or you get ahead and he is a tail,*2079

*then he wins and you have to pay him $5.00.*2085

*We are going to look at this from your perspective.*2088

*Let y be the amount you win, we want to find the probability distribution for this random variable.*2091

*I think of all the possible values that y can take and we want to find the probability of each one.*2099

*I’m going to start with just 0 because I like to include it.*2103

*In this case, there is no way that the random variable can come out to be exactly 0 here.*2107

*Depending on what the coins show up, either your friend is going to pay you or you are going to pay your friend.*2118

*There is no outcome that leads to a 0 exchange of money, the probability there is 0.*2124

*The probability that you are going to make $5.00.*2128

*To make $5.00, we have to get both coins being tails.*2136

*What is the probability of tail-tail and the chance of both coins showing tails is ½ × ½ is ¼.*2142

*That is your chance of making $5.00 out of this experiment.*2152

*The probability of you making $10.00 out of this experiment, for that to happen both flips have to be head.*2156

*You got to see 2 heads there.*2162

*The probability of 2 heads is ½ × ½ = ¼.*2166

*If the coins do not match then you have to pay your friend $5.00.*2171

*From your perspective, we got to win -$5.00 but you are really losing $5.00.*2176

*The way that happens is, if you show a tail and he flips a head, or if you flip a head and your friend flips a tail.*2186

*There are 2 possibilities there but each one of those has probability ¼.*2195

*¼ + ¼ is ½.*2201

*Those are all the different outcomes that can happen and those are all the different values of the random variable.*2209

*Those are all the different payoffs that can happen in this experiment.*2216

*I listed 0 even though it is not really a legitimate outcome here.*2219

*I just want to calculate it through, there is a 0 probability that no money is going to change hands here.*2223

*The probability of getting $5.00, we are looking at this from your perspective.*2230

*$5.00 win for you would happen if there are 2 tails.*2235

*To get 2 tails in a row or to get both your coins to come up tails, there is a ¼ chance of that.*2240

*To get 2 heads in a row which is what you need to make $10.00, there is also ¼ chance.*2246

*The probability of getting -$5.00 that means that your friend takes $5.00 from you, *2252

*that has to happen if you flip a head and he flips a tail, or vice versa.*2259

*Each one of those combinations has a 1/4 chance meaning that there is a ½ chance that you are going to end up paying him $5.00.*2265

*That is how we calculated each one of those probabilities for each one of those payoffs.*2275

*That is considered to be the probability distribution for this random variable.*2280

*That is the last example and that wraps up this lecture on random variables and probability distributions.*2287

*This is part of the larger probability series here on www.educator.com.*2295

*My name is Will Murray, thank you for watching, bye.*2299

2 answers

Last reply by: Dr. William Murray

Thu Sep 3, 2015 11:57 AM

Post by Hen McGibbons on August 29, 2015

at 17:20, you said there are n Choose r ways to choose y heads. why would you use n-r+1 choose r ways? because i thought this situation would be unordered, but with replacement. My reasoning is that after you choose a heads, you can put the heads back in the drawing and choose it again. but you said this situation is unordered and without replacement so i don't understand why.

1 answer

Last reply by: Dr. William Murray

Tue Sep 2, 2014 7:58 PM

Post by Ikze Cho on August 30, 2014

Hi

In Example 4 I didn't quite understand why we sometimes multiplied the probability of liverpool winning with 3 and sometimes not. In each case there were three matches, so in to me it would have made sense to multiply everything with three.

Could you please explain why my method is wrong?

Thank you