Probability 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 variable0017

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 40111

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 experiment0291

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 concerned0412

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 value0531

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 lecture0571

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 is0612

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 $10.000775

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 and0811

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 and0840

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 means1228

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 ⁺101283

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 roll1416

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 rolls1497

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 that1551

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 winning1716

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