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Post by Azhar Rahman on June 16, 2013

My lecturer gave the formula for Expected values as E(X)=µ=∑x.f(x)
with an x under the sum of symbol which i couldnt enter. Why is there a difference?

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Post by James Ulatowski on December 30, 2011

On The Random Variable lesson of Intro to Probability Distributions you said for the greater of two die that 6 is 11/36 "because you don't count 6 twice". As you know, you don't count any of the doubles twice, resulting in the odd number. Considering the sample space:
if you point out that these are simply the result of adding the rows and columns of the occurrence of each number -without double counting the double-it is a lot clearer as to how the resulting probabilities are obtained.

Introduction to Probability Distributions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Roadmap 0:08
    • Roadmap
  • Sampling vs. Probability 0:57
    • Sampling
    • Missing
    • What is Missing?
  • Insight: Probability Distributions 5:26
    • Insight: Probability Distributions
    • What is a Probability Distribution?
  • From Sample Spaces to Probability Distributions 8:44
    • Sample Space
    • Probability Distribution of the Sum of Two Die
  • The Random Variable 17:43
    • The Random Variable
  • Expected Value 21:52
    • Expected Value
  • Example 1: Probability Distributions 28:45
  • Example 2: Probability Distributions 35:30
  • Example 3: Probability Distributions 43:37
  • Example 4: Probability Distributions 47:20

Transcription: Introduction to Probability Distributions

Hi and welcome to www.educator .com. 0000

Today we are going to be talking about probability distributions. 0002

We are just going to start talking about them.0006

So far we have covered sampling method as well as a little bit about basics of fundamental of probability.0012

In probability distribution we are going to be playing those two ideas together and work on that so we can get what we want out of them.0019

Because of that we need to talk about what sampling and probability cannot do.0030

They cannot tell us something and because of that we need probability distribution and 0035

that is going to be the solution to what sampling and probability alone cannot accomplish.0040

With the probability distribution we are going to be talking about random variable and what that means and the expected value is.0047

We know a little bit about sampling and a little bit about probability, basically in sampling we know that 0061

what we are trying to do is estimate some variable by taking the sample of the population.0065

Here is the population of all the people we are interested in and we take a sample out and 0071

we look at those individuals to try and estimate what that variable is like in the population.0079

We know a little bit about the probability now in a particular population, we know the chance of getting a specific sample.0092

What is the probability of getting this particular sample.0105

You might want to ask yourself what is the relationship between these two things.0111

It is where we are getting the sample and we are looking at the mean and the standard deviation of some particular variable in order to estimate that variable.0116

In probability, what we are doing is it is not that we are actually taking a sample but we have a population.0130

We are trying to think about all the different kinds of samples, the sample space and then calculate the probability of the specific sample.0139

In one we are actually taking the sample and the other we are not actually taking a sample.0153

Here there are some limitations because even if we try to take a random sample, one that is not biased in some way.0159

Even though we try to take a sample we do not necessarily know how close that sample is to the population.0169

Here we have taken a sample but we know the probability of getting that particular sample.0176

We are still missing one piece of the puzzle, neither sampling nor probability helps us know the actual characteristics,0186

Things like shape and spread of the population.0194

We can know how likely this sample is given the population.0199

We could take our sample out and summarize that sample.0203

But we do not know how to understand the population if we do not already know what it is.0209

What do we do?0216

Here is where the probability distributions are going to come in.0219

In probability distributions, what we are going to do is take this population and look at the distribution of all the different possible samples and get their probabilities.0232

This way it is not that we are calculating one probability for one particular sample but we are calculating all the possible sample and all the probability that go along with it.0259

That is the probability distribution.0271

We are no longer dealing with single probability or single samples but we are looking at the universe of all the sample 0274

and the probability that go with them that might happen from a low population.0282

It is not necessary to know or we definitely know that this is a population but it could be a theorize population like a fair sighted point.0292

Those could be theoretical model.0309

That is the known population, we have sense of the population and from that we win from it, 0311

all the possible sample and all the possibilities that go with the sample.0321

Here is why we want to do that.0329

First we use probability, the fundamental principles of probability to figure out what samples from real populations look like.0332

We would want to do it from one sample at a time now we are going to do it for all the possible samples.0341

We figure out what the distribution of samples looks like then we take a sample from a different unknown population.0347

Here we have known population and now we get all samples and corresponding probability of those samples.0357

Then we have this unknown one, unknown population, and we take a sample.0377

What we do is we compare this sample to the results of the samples from the known population.0387

Why this sample like this is our theorize population?0394

Why is this sample unlikely in this universe of all possible samples from such a know population.0399

If this sample is highly like given that we know that this is probability then we could say these two guys are similar.0407

But if this sample is very unlikely then maybe we could say perhaps these two population are not similar.0417

This one is different from this known population.0428

That is the insight, we are going to use these probability distributions in order to help us figure out samples 0432

that come from population to try and figure out what the unknown population is like.0440

It helps us to find what a probability distribution is.0450

These are all the samples that can be drawn from an a known population and their corresponding probabilities.0456

When you think about probability distributions, I also want you to think of what it is called a sampling distribution.0489

What it is really is, it is not a distribution of a single entity or people or companies.0502

It is not a distribution of single data point.0511

Instead it is a sampling distribution.0515

It is a distribution made up of a whole bunch of samples.0519

Now that we know what probability distributions are, for and why we need them so desperately.0529

Let us try to smoothen out the road from probability to probability distributions.0535

In probability we learn about sample spaces.0546

Remember when we are talking about sample spaces, we are talking about all the different possible samples that we could have.0550

When we talk about flipping two coins and we look at what is the probability of getting 2 heads in a row?0557

We created these sample spaces of all the possible different outcomes.0565

For instance, if we look at the sample space for 2 dice we might look at something like this, dice 1, 2.0579

We might have 1, 2, 3, 4, 5, 6 and 1, 2, 3, 4, 5.0597

If we look at the sample space that would be 1-1, 1-2, 1-3, 1-4, 1-5, 1-6.0610

Or it might be 2-6, 3-6, 4-6, 5-6, 6-6.0622

If you fill in all the different possible outcomes there are 36 possible outcomes.0632

If you fill in the rest of this table you could see the entire sample space.0642

Each of these roll of a dice presuming that this dice are fair, each of these we have a probability of 1 out of 36.0647

The likelihood of this out of all the possible outcomes is 36.0659

That would be looking at the sample space.0668

That is the probabilities of all the different outcomes.0672

When we talk about probability distributions, often we are talking about just one number and the distribution around that one variable, whatever that variable is.0677

Here notice that we have 2 numbers but we may want something like the sum of 2 dice.0690

Here the sum would be 2 and the sum would be 3 and the sum would be 4.0699

Maybe we want the probability distribution that looks something like this.0706

Here I will put all the different possible sums.0713

I may not have room for this so I might draw them separately.0717

For instance, can I have a sum of 0?0720

No, I cannot have a sum of 1 either.0724

If I roll 2 dice the lowest possible sum I could have is 2, a 1 and a 1.0727

I could have a sum of 2, 3, 4, 5, 6, 7 but I could also have, I will continue this on this side.0733

It will also have sums of 8, 9, 10, 11, 12.0747

I cannot have any sum higher than 12 because the 6 and 6 on each dice is the highest possible sum I could have.0752

My probability distribution would be what is the probability of the sum of 2?0764

That would be only one outcome out of all 36 outcomes would give you that sum of 2.0773

That would be 1 out of 36.0783

The same thing with 12, the only possible combination that I could get a 12 is this 6 and 6 right there.0792

That would be also 1 out of 36.0800

Now let us think about 3, how many rolls of the dice could possibly add up to 3?0803

1 and 2 but also 2 and 1.0813

Any numbers higher than that is not going to work.0818

Here we would write 2/36 because 2 different rolls could give us the same sum.0823

Notice that not all of the probabilities are equal.0833

This one is less probable and this is more probable.0836

What about 4?0843

For 4, now we are starting to get into some bigger numbers here.0844

1 and 3 will work but so is 2 and 2, as well as 3 and 1.0849

That is going to be 3/36.0859

Notice the pattern here so far?0863

Also notice the pattern here is going like this and like this.0866

Let us continue that pattern and see if it is correct.0872

Our prediction would be that the sum for 5 would be 4/36 and let us see if that is true.0876

If the sums combine and so thus 4 and 1.0883

Let us see if these diagonals also do.0891

These diagonals would be 2 and 3, as well as 3 and 2.0893

Again, we know we could see our rule generalizing here.0899

That is going to be 4/36.0904

Actually this ends up working all the way up to 7.0910

As 7, this would be going to be the longest diagonal.0917

I will that in right here, 6 and 1, but also 5 and 2, 4 and 3, 3 and 4, 2 and 5, and also 1 and 6.0922

All 6 of these here on the diagonal will add up to a sum of 7.0949

That is why in games like stocks 7 is the highest probability rule.0957

Is another game where 7 is the highest probability roll that is why it is not allowed.0964

Here, you will see that what happens after these diagonal is that it becomes smaller again but in a perfectly symmetrical way.0976

What goes up? 1, 2, 3, 4, 5, 6 comes down 5, 4, 3, 2, 1.0985

That is because you are just finding those diagonals but now it is going from big to smaller and smaller then you will end up here the 1/36.1000

This is probability distributions for us to start with because it is easy for us to see how we got these probabilities from these sums.1013

This is a relatively small sample space and a small probability distribution we could fill in.1026

There are going to be bigger spaces lined up each knot necessarily calculated by looking at the different combination and1033

we will look at some of the algorithms and theorems that have been developed in order that we could have shortcuts so that we do not have to look at the entire sample space.1041

But I want you to know where this comes from.1054

It all comes from the sample space and looking at the probabilities.1056

In the previous probability distribution we looked at the sum of the probability distributions of two dice.1065

In that case, the random variable of one single variable that we are interested in on this probability distribution is the sum of the two dice.1075

That is one example of a random variable.1085

It is the thinking here, the thing that you are finding the probability of, the one variable.1088

It does not have to be sum, it could be something else.1099

For instance, in other games you might be interested in the probability distribution of the greater of the 2 dice.1103

Here we have another probability distribution where instead of the sum I am looking for the probabilities of those sums.1110

We choose the greater number and the probability of the greater number.1119

Here I have just rewritten what I have written on the previous page except nice and neatly.1130

One thing I want you to notice is if you add up all of these probabilities they add up to 1.1135

That is good because this shows me that we have covered the entire scan of the different outcomes as well as the different probabilities.1141

There is no part of the sample space that have not has been untouched.1155

You have touched all of it.1159

Here we see once again as the sums go up 7 will be the highest probability roll of the highest probability combination.1162

It starts going down after 7.1174

2 and 1 is perfectly symmetrical so that these sides are just the same and 7 is the mirror point.1178

If we are looking at we roll some dice and we are looking at what is the probability that the greatest number there out of the two is 1?1189

There is only one case of that, when it is 1 and 1 the greater number is 1.1198

That is 1/36.1207

But notice that this does not goes up and down because here what is the probability that when you roll a dice that the greatest number there will be 2?1210

That is only going to happen in case of 1 and 2, 2 and 1, 2 and 2.1221

That is 3/36.1230

Let us get down to 6.1233

6 is the highest number so it is going to be the greatest number at the time where all of the combinations that are like 6 and 1, 6 and 2, 6 and 3, 6 and 4.1235

As well as 1 and 6, 2 and 6, 3 and 6.1248

6 is frequently the highest number but we do not count 6 and 6 twice which is why it is 11/36.1255

Even so we add all of the probabilities we have a total of 1 and that shows me we have covered the entire space.1262

These are samples of probability distributions of two different random variables.1271

One is the random variable of sum and the other is the random variable of greater number.1276

We could have a random variable like which is the product, two numbers multiplied together.1284

We could have all sorts of different random variables that we are interested in.1298

The fact is that we think upon which the probability distribution is being decided.1301

That is the thing you want to find the probabilities of.1308

Now we know the variables and now we know the probability distributions, what is expected value?1315

Here is the thing, once you a probability distribution like this, this is that rolling two dice and what is the probability of the sums?1321

Here once you have this probability distribution of the sums, that is our random variable, and we often call our random variable X.1332

We could also call it I or M or Y but we call it big letters.1342

It would be handy to know the mean of this probability distribution because every distribution has a mean.1351

It would be nice to know what the mean of this distribution is.1360

That is often called the expected value, the mean of the probability distribution or the mean of the sampling distributions.1365

The reason it is called the expected value is this.1375

Over time if you keep rolling two dice over and over again, this probability should emerge, the law of large numbers.1378

Over 10,000,000 rolls of two dice, we should see the frequencies corresponding to this probability.1389

It would be nice to know what we should expect even without rolling a dice 10,000,000 times.1401

It would be nice to know what the mean would be on average.1410

In order to find the expected value, what we are doing here is what we did average because these two is not quite as frequent as the 7.1419

These two should not count as much as the 7.1431

This 12 although it is a bigger number it should not count as much as 7 because it is only going to have a small portion of a time.1433

What we are doing is we are going to not only pick the sum and put it in our estimate but we are going to weight it by how likely we need it.1441

If it is very unlikely it only contributes a little bit but it is very likely then it contributes a lot to the expected value.1455

Out of those contributions you should be able to see what is the expected value over time?1475

What is going to be the most likely value of this random variable?1483

In order to find the expected value, you will often see it notated like this, the expected value of x, this random variable.1492

Another notation for this is mu sub X.1505

The reason why we think of it as a mu is that its probability distributions are theoretical distributions.1512

Remember, theoretical distributions are more like populations than samples.1518

It is mu sub X, it is not just mu.1524

If it is only mu it would be the population.1528

When you see mu sub X this means this is the expected value.1531

This can be easily found by taking the sum from i(whatever your x are).1538

In this case this is my x because the sum is my random variable × how likely it is.1551

We need to do this for however many x we have.1569

Here what we see is that if x contributes a lot because the probability is very high then this is going to be a bigger number.1573

If x only contributes a little bit its value is going to be a little bit diminished.1596

When we add all of those up, we hear the course of who is the loudest?1602

That is our expected value.1609

Some people contribute a lot and some people contribute a little bit when we add all of them up what we find is the true story.1612

In order to find this, it is going to be helpful to use Excel.1622

If you want to download the available Excel file I have put in all the sums as well as the probability.1625

I have just put in 1/36 but Excel will put it in decimal form for you.1638

What we want to do is you want to think of this multiply together as the contribution of how much is it contributing to the expected value?1645

In order to do that, we would take this x and multiply it by the probability of the x.1654

This 2 contributes a little bit but the 7 contributes quite a bit.1664

It is much bigger than this 2.1675

It goes down again where the 12 only contributes a little bit.1681

In order to find expected value we add all of these up.1686

We add all of these contributions up.1690

Let us see what is left.1693

Our expected value or the mu sub X is 7.1696

Over many rolls of the dice what we will see is on average the expected value is 7, that is the mean of this probability distribution.1701

That expected value.1723

Now let us move on to some examples.1727

You have all the tools of the game.1731

In example 1, at the state fair you could play a fish for cash, a game of chance that costs $1 to play.1733

You blind the fish out of cart that has a dollar amount that you have won from a giant fish bowl.1739

The games have these probabilities of winning posted on the wall.1745

Is it worth playing this game?1749

Here we the winnings that you could potentially earn.1751

You could win from $1 all the way up to $900.1755

Here are your chances of winning.1759

You have 1 out of 10 chance of earning back your $1 and you have 1/120,000 of winning $900.1761

You are getting a lot of bang for your buck.1774

Is it worth playing this game?1777

I have put all of this information on example 1 tab, these are the winnings and I just put them in here and Excel has changed it in decimal form.1781

The first thing you want to do is check whether all of this probability actually add up to 1.1796

Remember these are posted on the wall.1803

Even out there they are telling you the whole story.1806

Let us sum these probabilities up and make sure that we cover the entire space of winnings and probability of those winnings.1810

These probabilities only cover 22% of that probability space and there is the probability of 1 a 100% total.1824

Let us think.1835

What must be missing?1837

I think the game probably does not advertise that you count the probability of winning that game.1842

There is that probability and they are probably not just telling you.1848

Let us see.1852

I am going to take all of that and add that over here.1859

In this row I am going to put 0 as the potential winning and the probability of getting 0 should be 1 – the sum of all the rest of the possible outcomes.1871

That is almost 78%.1894

You have likely to win nothing but once we do this, if we add up all of these including the 0 now we should have a total of 1.1898

That is good and we want that.1911

This shows us this is now a complete probability distribution of the random variable winnings.1914

What is the contribution of each of these winnings to this expected value?1925

Over many plays of this game, what is the average winning?1931

Let us look at each one contribution and then multiply that out and we sum this up.1941

When we look at this sum here, you will see $.60 as the expected value.1959

Even though there is this chance of winning $900 as well as $.60.1965

It turned out that on average if you play the game over and over again, the average winning is going to be about $.60.1970

You can win $.60 on average so the expected value of winnings or you can write it as mu (w) is $.60.1982

Is it worth it playing this game?2001

That costs you a buck to play the game so if you play the game over and over again, let us say a 100 times that is going to cost you $100.2005

But over time if you multiply this by a 100 you are going to win $60 for every $100 you spend.2016

That is not worth it.2027

Over the long call you are going to be losing money.2030

It does not matter that much if you are just going to play the game once.2033

It is not going to tell you whether that part that you picked is going to be $.60 because remember there is no possible way you could earn $.60.2043

This is not what that means, probably it is going to be 0 because it is 79%.2053

4 out of 5 times you are going to be drawing 0.2060

It is not about any 1 particular turn.2064

You might think that is useless.2069

Actually it is not.2073

Move yourself around and put yourself in the seat of the guy who owns this game, fish for catch.2075

You want to know the expected value.2083

You are the owner of the game.2085

You want to know on average, all of these different people are going to play are you going to be losing money at the end of the day or you are going to be grateful you did not?2088

What the order of this game would say is this is a good game because if people play even though I will lose $.60 for every $1 roughly on average, I will be gaining $.40.2097

That is my profit.2113

This is not really for just single people.2116

We are not trying to predict single events but we are trying to predict events over time and over many examples.2122

Here is example 2.2131

According to our recent government report only 16% of occupants in trucks wear seat belts, supposed you randomly sample 3 occupants of trucks 2134

what is the random variable and which doubt that this 16% estimate if not of the 3 is wearing seat belts?2146

Let us see.2155

In order to create a probability distribution we want to decide on what the random variable should be.2157

We eventually want to know what if none of the 3 wear seat belts?2164

Maybe we want to know how many seat belts and passengers?2170

Seat belt and passengers.2173

We have 0 seat belts and passengers that is our none of the 3.2179

We could have 1, 2, or 3.2183

We also want to know what is the probability of these seat belted passengers?2185

Now we want to figure this out.2198

If this was like head or tails, all of these combinations we have equal probability but 2201

this all have equal probability because only 16% of occupants of trucks say they wear seat belts.2209

Although it is useful to have the sample space this is not going to be enough.2216

First let us just look at the sample space.2227

Here is 1, 2, and 3, and I will put in s for seat belt and n for no seat belts.2230

Half of these are seat belted and half are not.2245

Here is our sample space and we see that the 3 people all wear seat belts and that is 1/8 of the sample space.2261

This is also 1/8 of the entire sample space.2273

The story is not that simple because that would only be if wearing a seat belt and not wearing a seat belt or equally likely.2278

That is not the case.2286

These are independent events so we could use the multiplication rule.2289

Let us say we have these 3 spaces what is the probability that this guy here is not wearing a seat belt?2296

That is going to be 84%.2302

16% wearing a seat belt and the other side of that point is 84%.2304

That is the probability for each of these seat belts because think of them as independent events.2316

For number 3 where all 3 people are nice seat belt wearing, well abiding citizens in this truck that is going to be 163.2324

These are a little more complicated.2344

Here let us think about this, what is the probability that one person is wearing a seat belt?2346

This got one of the seats but these people are not.2352

This is not much more likely also in the sample space.2364

There are 3 of these.2370

Here is one, here is another one, there is another one.2372

As we are using the addition rule, we would add these 3 times but we could also just multiply it by 3 to make it easier.2380

What about if 2 people are wearing seat belts and 1 person is not?2391

That would be .16 for the 2 people wearing seat belt that is the probability that 2396

2 people are wearing seat belts × .84 the probability that they would not be wearing seat belts.2407

Here we see that there are also 3 cases.2416

We could get the addition rule and add these 3 times but we could just multiply it by 3.2423

Once we do that then we could see the actual probability.2431

Here we do not need to multiply by anything because there is only 1.2436

It is like multiplying by 1.2440

I am just going to calculate this in my Excel file so I could put in .843 so that is .84 × .84 × .84.2446

That is 59.27%.2466

This should be .5927.2479

We could do the other one just to finish it up 2 × .1, 3 × .16 × .84 × .84.2488

The next one is 3 × .16 × .16 × .84.2510

The last one is .163.2523

We see that it is very likely where all 3 of them are not wearing a seat belt.2528

It is less likely than this but still pretty likely about 34% of the time that your sample have 1% wearing seat belts.2538

Less likely that two people would be wearing seat belt that is only like 6%.2551

If there is less than 1% chance that all 3 people will be wearing a seat belt.2555

Would you tend to doubt the 16% estimate if none of the 3 were wearing a seat belt?2581

We know that if we took a sample 59% at the time that sample will have 0% wearing seat belts.2587

It is likely that sample is consistent with this estimate of 16% of occupants in trucks do not wear seat belts.2599

We would not necessarily doubt that 16% occupants.2609

We might still be wrong but we do not have reason to doubt it.2612

Here is example 3, Apple is going to get you to buy the Apple care warranty for $250 for your laptop.2620

If you buy this Apple care thing you will get unlimited number of free repairs but if you do not you must pay $150 per repair.2627

You find this information below on the web.2642

What is the random variable here and according to this data is this worth getting the warranty?2646

Here you want to predict how likely it is going to be that I am going to need a bunch of repairs?2653

If I need only 1 repair or 0 repair the I probably should not buy this warranty thing but if I am going to need to get repair frequently then I probably want to buy the warranty.2660

Here we could see the probability of the repairs listed here and here are the repairs.2675

This probability distribution has the random variable of repairs.2682

According to this data is it worth getting this warranty?2688

One thing we want to do is to look and see what the expected value is.2693

On average, over million people are buying these laptops what is likely the value of repairs.2699

How many repairs are we going to need on average?2710

I have put this data on this table here in example 3, it is the same table.2714

It would be nice to find the expected value of x.2725

In this case x is repairs.2731

In order to find that, we want to find the contribution of each value of this random variable.2735

It is this times how likely it is.2749

0 contributes more than 4 because it is much more frequent that people need only 0 repairs.2753

Then we add these all up.2769

Remember our expected value is repairs.2773

We on average will expect about 1.1 repairs.2777

No one person will need 1.1 repairs they might need 1 repair or 2.2784

On average, it will average up on 1.1.2790

Remember this is on repairs, we do not know how much it is in dollar amounts.2794

If we want it to know in dollar amount, we know that each repair if we identify the warranty will cost $150.2798

That will be just this times $150.2807

On average we would expect to spend $155 is that going to make Apple care worth it?2812

I will say according to this data it is highly likely that you will need an average 1 repair.2822

It is probably not worth getting Apple care.2832

Here is example 4, there are 5 television shows you want to watch before your class that will begin 74 minutes later.2842

If you randomly pick 2 shows to watch out of this 5, what is the probability that you will have finish watching both shows before your class?2849

It is helpful here to keep in line the questions even though it is not asked here.2859

What is the random variable?2863

Do not be tricked because here it shows the shows A, B, C, D, and E, and it shows the minutes how long each shows is.2866

It does not show you the probability distribution at all.2881

In fact it is not like that there is a probability distribution for A.2886

How like is A?2890

Equally like as B, C, and D if we just pick randomly.2892

What we want to know is how many minutes 2 shows will be when added together?2896

Picking 2 shows and getting the sum of those 2 shows.2905

Eventually what we want to have is sum of the length of 2 shows.2912

We want all the different combinations likely AB, AC, AD, and all their sums and then the probabilities of those sums.2926

That is going to take more room than what we have here so if you pull out this Excel file and go to example 4.2939

I have put this information on here.2952

What you want to do is first start off by just looking at the sample space.2955

Let us start with the sample space.2960

The sample space might look something like this.2964

First we want all of the combinations with A as the first show we picked.2967

B will be the second show.2976

C will be the second show.2980

Here we have D as the second show.2983

Maybe it is helpful if we label these.2985

First show and then first show × second show.2986

We have all these different things.2999

Later what we can do is get the sum by adding this and this.3002

We could do that later.3011

Let us create the sample space.3013

The next combination should be all the combinations with B as the first show and A will be the second show.3015

C, D, E as the second show.3032

Next we have all the combinations where C as the first show and then A and B are the second show D and E are the second show.3036

Let us go down to D being the first show and you could have A or B as the first show.3055

C as the first show, A and B as the second show, C as the second show, or E as the second show.3066

Presumably you do not want to watch A after you just watched A.3074

Finally, we have E as the first show, A, B, C, D as the second show.3082

We have all these different shows.3094

All the combinations and we could quite easily calculate the sums.3097

These are all the sum total minutes.3107

What we want to do is sort this by sum because eventually we want a probability distribution that will use sum.3111

Here I wanted to sort by sum because eventually we want to have probability distribution that have just these sum on it.3135

We do not care if 63 was created by watching B and D or D and B.3145

We just care that it adds up to 63 minutes.3151

I am going to go over here and start my probability distribution.3158

Here I will put the sum and I have a couple of different sums here.3163

For instance I have 63 that is the sum that come up, 65 comes up, 68 comes up, 70, 71, 73, 75, 78.3168

I want to know how likely it is 63 given all of these potential different combinations.3190

These are all equally probable because each show has 1 out of 5 chance of being picked as the first show and 1 out of 4 chance being picked as the second show.3198

Here what we will do is we need a formula that we could put countif and we could say in this range we could lock it down and it is not going to change for us.3220

In this range countif is it is 63.3235

Let us put all of that over however many different combinations there are.3240

That is all of this.3247

I think there is actually 20 combinations.3251

In this way we can have Excel do the work for us.3253

I am locking that down and once I do that it says that 2 out of 20 that ends up being 1 out of 10 which is .1.3258

What we can do is just copy and paste this all the way down.3271

We find that most of these are equally probable like 63, 65 minutes as the sum but 68 minutes is twice as likely because it is 4 out of 20.3275

Twice as likely as this 70 minutes or 71 minutes.3291

Here I will put probability of these particular sums.3297

What ends up being helpful is to look back at the questions and say now that we have what we want to know here?3303

If you randomly pick 2 shows to watch what is the probability that you will have finish watching both shows before your class.3311

Let us say you just arrive at your class you could use all 74 minutes to watch.3318

The cut off would be around here.3324

We could watch all these different combinations and still be under 74 minutes.3326

These are the only combinations that are not under 75 minutes.3334

The way that we would write this in probable notation is p where x is the sum in this case is less than 74 minutes.3339

You could just add these probabilities up using the addition rule.3359

You could just add these probabilities up.3365

In order to get that and we could put the sums and that should give us 80% because 3369

the only 2 we are leaving out are these and each of those have a probability of 10%.3382

If you randomly pick shows to watch there is 80% probability that you will finish watching both of those shows before you go on to your class.3387

Nice and helpful of probability to help this figure out our TV watching schedule.3397

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