# Money Management

## Renting and Homeownership

### Learning Objectives

After completing this section, you should be able to:
1. Evaluate advantages and disadvantages of renting.
2. Evaluate advantages and disadvantages of homeownership.
3. Calculate the monthly payment for a mortgage and related interest cost.
4. Read and interpret an amortization schedule.
5. Solve application problems involving affordability of a mortgage.
After renting an apartment for 10 years, you realize that it may be time to purchase a home. Your job is stable, and you could use more space. It is time to investigate becoming a homeowner. What are the things that you must consider, and what is the financial benefit of owning as opposed to renting? This section is about the advantages, disadvantages, and costs of homeownership as opposed to renting.

### Advantages and Disadvantages of Renting

When renting, you will likely sign a lease, which is a contract between a renter and a landlord. A landlord is the person or company that owns property that is rented. The lease will detail your responsibilities, restrictions on activities, deposits, fees, maintenance, repairs, and rent during the term of the lease. It also defines what your landlord can, and cannot, do with the property while you occupy the property.
Like leasing a car, there are advantages to renting but also some disadvantages. Some advantages are:
1. Lower cost.
2. Short-term commitment.
3. Little to no maintenance cost. The landlord pays for or performs most maintenance.
4. You need not stay at end of lease. Once the lease term is over (the lease is up), you are not obligated to stay.
5. If renting in an apartment complex, there may be a pool, gym, or community room for renters to use.
Of course, there are disadvantage too:
1. No tax incentives.
2. Housing cost is not fixed. When the lease is up, the rent can change.
3. No equity. When you are done living in a rental, you have built no value.
4. Restrictions on occupants. There may be a limit on how many can live in the apartment.
5. Restrictions on decorating. The property is not yours, so any decorating or improvements need landlord permission.
6. Limits on pets. Permission for pets, and their number and type, will be set forth in the lease.
7. May not be able to remain when lease term is over. The landlord can, at the end of your lease, invite you to leave.
8. The building may be sold, and the new landlord may institute changes to the lease when the previous lease expires.
Renting has fees to be paid at the start of the lease. Typically, when you rent, you will pay first and last months’ rent and a security deposit. A security deposit is a sum of money that the landlord holds until the renter leaves the rental property. The deposit will cover repairs for damage to the apartment during the renter’s stay but may be returned if the apartment is in good condition. If your landlord runs a credit check on you, the landlord may charge you for that.

### Advantages of Buying a Home

The advantages to buying a home mirror the disadvantages of renting, and the disadvantages of home ownership mirror the advantages of renting.
Some advantages to buying a home are:
1. There are tax incentives. The interest you pay for your mortgage (more on that later) is deductible on your federal income tax.
2. There are no restrictions on pets or occupants, unless laws in your area specify limits for homes.
3. You can redecorate any way you wish, limited only by the laws in your area.
4. Once your mortgage is set with a fixed-interest rate, your housing cost is fixed.
5. Your home grows equity, that is, the difference between what you owe and what the house is worth grows. You can use the equity to secure loans, and you recover the equity (and more if you’re fortunate) when you sell the house.
6. As long as you pay your mortgage and maintain the home to the standards of your community, you can stay as long as you wish.
Some disadvantages to home ownership are:
1. The cost is higher than renting. Mortgages and associated costs are typically higher than rent for a similar living space.
2. The owner is responsible for upkeep, maintenance, and repairs. These can be extremely costly.
3. The owner cannot walk away from the property. It can be sold, but simply leaving the property, especially if not paid off yet, has serious consequences.
The big question of affordability looms large over the decision to rent or buy. Renting, strictly from an affordability viewpoint, comes with much less initial outlay and smaller commitment. If you do not have sufficient income to regularly save for possibly expensive repairs, or your credit isn’t quite as good as it needs to be, then renting may be the best choice. Of course, even if you can afford to buy a home, you may choose to rent based on the comparative advantages.
Buying a home really involves two buyers. You and the mortgage company. The mortgage company has interest in the home, as they are providing the funds for the home. They want to protect their investment, and many fees are about the bank as much as the buyer. They fund a mortgage based on the value they assign the property. Not you. This means they will want some certainty that the home is sound, and you are a good investment.
In the end, you must weigh your options and carefully consider your priorities in choosing to rent or buy a home.

### Mortgages

Some people will purchase a home or condo with cash, but the majority of people will apply for a mortgage. A mortgage is a long-term loan and the property itself is the security. The bank decides the minimum down payment (with your input), the payment schedule, the duration of the loan, whether the loan can be assumed by another party, and the penalty for late payments. The title of the home belongs to the bank.
Since a mortgage is a loan, everything about loans from The Basics of Loans holds true, including the formula for the payments.

### Monthly Mortgage Payments

The formula to calculate your monthly payments of principal and interest uses APR as the annual interest rate.
To find the total amount of your payments over the life of the loan, multiply your monthly payments by the number of payments.
To find the total amount of your payments over the life of the loan, multiply your monthly payments by the number of payments. This can be useful information, but not too many people reach the end of their mortgage. They tend to move before the mortgage is paid off.
With the principal of the mortgage and how much total is paid over the life of the mortgage, the cost of financing can be found by subtracting the principal of the mortgage from the total paid over the life of the mortgage.

### Reading and Interpreting Amortization Tables

Amortization tables were addressed in The Basics of Loans. They are most frequently encountered when analyzing mortgages.
The amortization table for a 30-year mortgage is quite long, containing 360 rows. A full table will not be reproduced here. We can, though, read information from a portion of an amortization table.

### Escrow Payments

The last few examples have looked at mortgage payments, which cover the principal and interest. However, when you take out a mortgage, the payment is sometimes much higher than that. This is because your mortgage company also has you pay into an escrow account, which is a savings account maintained by the mortgage company.
Your insurance payments will be set by your insurer and the mortgage company will pay them on time for you from your escrow account. Your property taxes are set by where you live and are typically a percentage of your property’s assessed value. The assessed value is the estimation of the value of your home and does not necessary reflect the purchase or resale value of the home. Your property taxes will also be paid on time by the mortgage company from your escrow account.
For example, in Kalamazoo, Michigan, the effective tax rate for property is 1.69% of the assessed value of the home. These escrow payments, which cover bills for the home, can increase the monthly payments for your home well beyond the basic principal and interest payment.

### Check Your Understanding

### Key Terms

1. Lease
2. Landlord
3. Mortgage
4. Escrow account
5. Assessed value

### Key Concepts

1. There are many points of comparison between renting and buying a house.
2. Before deciding to buy a house, you should carefully consider all the responsibilities that come with home ownership.
3. Renting comes with more restrictions on the renter, but with fewer costs and is easier to move from.
4. Owning a house has more costs but has more freedom, plus the owner creates equity.
5. Mortgages are loans, and payments are calculated in the same way as any other loan.
6. Amortization tables help a homeowner understand the mortgage and how the payments are applied to the principal and interest.
7. In addition to paying the amount financed for a mortgage, the monthly payment will include an escrow payment, which covers insurance and taxes.

### Video

1. Rent or Buy
2. Closing Costs

### Formula


# Money Management

## Income Tax

### Learning Objectives

After completing this section, you should be able to:
1. Determine gross, adjusted gross, and taxable income.
2. Apply exemptions, deductions, and credits to basic income tax calculations.
3. Compute FICA tax.
4. Solve tax application problems for working students.
Before the start of the American Civil War in 1861, most of the country’s revenue came from tariffs on trade and excise taxes. However, this fell far short of the high cost of the war. Because of this, the federal government enacted the nation’s first income tax with the Revenue Act of 1861, which created the Internal Revenue Service as we know it today.
No one likes paying income tax, but it is a reality of life. In this section, we will learn about Form 1040, the U.S. Individual Income Tax Return, and ways to prepare for tax time.
The U.S. tax code may change from year to year. Because of this, this section includes examples of how taxes, deductions, and exemptions might be computed. The types of income, deductions, and exemptions that are used in the examples are used in the current tax code.

### Gross, Adjusted Gross, and Taxable Income

Your income drives how much you pay in taxes. The more you earn, the more you are likely to pay. But your income alone is not the full story. When you add all the money you earned from your job, freelance work, interest from savings, and other sources, you have your gross income. If you are an employee, your income from your job will be reported on a W-2, which is sent to you by your employer. Income from freelance work will be reported on a 1099-MISC form, and is sent by the company that paid you. Income from interest is reported on a 1099-INT form and comes from the entity that paid the interest.
Before you determine how much you owe in taxes, you will make certain adjustments to that gross income. You will deduct, or subtract, some of income from the gross income. That’s your adjusted gross income, or AGI. That is still not what you are taxed on. Next, you need to apply exemptions to your income. These are pieces of income that the government does not tax. After that is done, you reach your taxable income. We will look at each of these parts of the taxable income.
You will notice that your paycheck already has taxes taken out of it. Your employer will withhold some of your income, sending it directly to the federal, state, and local governments. It is an estimate of how much you will owe in income tax. In the end, it reduces how much you will pay when your taxes are due. If they withhold too much income, you will receive the extra they withheld in the form of a refund.
Your adjusted gross income (AGI) is computed before your taxes are determined. It begins with the gross income, and then subtracts from that income any deduction. Deductions are expenditures on your part that the government won’t tax. These deductions include money deposited into tax-deferred investments, and mortgage interest that you paid, charitable contributions if you made any, medical bills over a threshold, medical insurance under certain circumstances, and property taxes. If you add all these up, and they are all legal deductions, the sum is subtracted from your gross income, leaving the AGI.
Remember that your AGI is not your taxable income. Exemptions need to be subtracted from the AGI to reach your taxable income. Exemptions are income that the government does not tax. Some examples of exempt income are disbursements from health savings accounts for qualified medical expenses, bond interest, some IRA distributions, and gifts given that are under $16,000. Note that exemptions are different from deductions: exemptions are excused incomes, whereas deductions are excused expenditures.

### Tax Credits

Another piece of the tax puzzle is tax credits. This is money subtracted from the tax you owe.
Tax credits are very different from deductions or exemptions. Deductions and exemptions are taken away from your gross income before the tax you owe is calculated. A tax credit, is subtracted, dollar for dollar, from your tax bill. Once the tax you owe is calculated, subtract the any tax credits from that calculated tax.
Some of the tax credits are refundable. This means that if subtracting them from your tax results in a negative number, you receive a tax refund. For more details, see this article about tax credits.
The federal government has placed income limits and restrictions and on those eligible to receive tax credits because their value is so high. Here is a partial list of tax credits that you might qualify for:
1. Earned income credit is a refundable tax credit for low- to moderate-income workers and ranges from $560 to $6,935 depending on dependents and income. This is refundable
2. American opportunity credit is a credit taken by parents who have children enrolled in college at least half time and pursuing a degree. This credit is worth $2,500 per student for the first 4 years of undergraduate school, subject to income limits. This is a refundable tax credit.
3. Lifetime learning credit is a credit is equivalent to 20% of educational expenses, up to $2,000 per year, subject to income limits. There is no cap to how many years you can apply for this credit.
4. Child tax credit is worth $2,000 per child under the age of 17 if that child lives at home at least half the year, subject to income limits. This is a refundable tax credit.
5. Child and dependent care tax credit was designed to help pay for child care while the parent works. The amount of the credit is dependent on your income. However, the maximum amount that can be received is, in 2022, $4,000 for one eligible person, or $8,000 for two or more qualifying people. A dependent qualifies if they are a child under 13 years old, a spouse who is unable to care for themselves, or some other qualifying person. This is a refundable tax credit.
6. Premium tax credit was created by the Affordable Care Act, and it is one that is received by many people throughout the year. In essence it is a health insurance premium subsidy. The amount of the credit is based on your income and the price of health insurance in your area. This is a refundable tax credit.

### Computing FICA Taxes

FICA stands for the Federal Insurance Contributions Act of 1935. FICA taxes are used solely to fund Social Security and Medicare and are separate from federal income tax. It amounts to 7.65% of your gross pay, which is withheld from your paycheck automatically. Your employer is required to match the 7.65% amount. Of the 7.65%, 6.2% goes to Social Security (SSI), and 1.45% goes to Medicare.
As of 2022, SSI tax only applies to the first $147,000 of earnings. Any gross income above that is not taxed for social security. This limit changes every year.
Medicare tax, on the other hand, applies to the entirety of your gross income.

### Calculating Your Income Tax

Your income tax bill and your income tax rate are based on your taxable income. The tax system in the United States is progressive, meaning that the tax rates are marginal so the higher your taxable income the higher the tax rate you will pay. Taxable income is broken into brackets, or ranges of income. Each bracket has a different tax rate. The tax brackets and rates for single filers as of 2022 are given below:
So if your taxable income is $76,500 and you are filing as a single filer, your tax bill will be 22% of that $76,500, right?
Wrong. Your income is split among those brackets and the money in each bracket is taxed at that bracket’s tax rate. Seems confusing. Here is a list of steps to follow to find the tax owed.
Step 1: Find the bracket for the income.
Step 2: For each bracket below the income bracket, the tax from that bracket is:
Step 2a: Find the difference between the upper limit of that bracket and upper limit of the next lower bracket. If this is bracket 1, use 0 as the upper limit of the previous bracket.
Step 2b: The tax from that bracket is the bracket tax rate applied to the difference from Step 2a.
Step 3: For the bracket that the income belongs to, find the income minus the lower limit for the bracket.
Step 4: The tax for the bracket of the income is tax rate for that bracket applied to the difference found in Step 3.
Step 5: Add these various tax values to get the total income tax.
There are various tax brackets, and the rates may change in any given year. The income limits may also change. For all examples going forward, we will use the single filer tax brackets, even if those brackets are not appropriate (e.g., married or head of household filers).

### Check Your Understanding

### Key Terms

1. Gross income
2. Adjusted gross income
3. Exemption
4. Taxable income
5. Deduction
6. Tax credit
7. Earned income credit
8. American opportunity credit
9. Lifetime learning credit
10. Child tax credit
11. Child and dependent care tax credit
12. Premium tax credit

### Key Concepts

1. Federal income tax is based on income after certain adjustments.
2. Gross income is income from all sources, including gifts and winnings.
3. Before taxes are calculated, the taxable income is found by subtracting deductions and exemptions from gross income.
4. Income tax is progressive, increasing in rate as income increases.
5. Being in the 32% tax bracket means some of your income is taxed at 10%, some at 12%, some at 22%, some at 24%, and the rest at 32%.
6. Income in each tax bracket is taxed at that bracket’s rate, which means in 2022 the first $10,275 earned is taxed at 10% only.
7. Tax credits are subtracted from the taxes that are owed.
8. Some tax credits are refundable, which means they can make the amount you owe negative, which results in a refund.

### Video

1. Deductions Versus Credits
2. Tax Brackets

### Project

### Creating Your Future Budget

In this project, you will create a budget based on a job you are likely to have after you graduate.
1. Go online and research the average starting salary for the profession you are studying for. Use at least two sources. Be sure to record the web address from your search.
2. Approximate your monthly take-home pay. You may use the SmartAsset website to estimate this.
3. Use the 50-30-20 budget philosophy to determine how much you should budget for needs, wants and savings, or extra debt reduction.
4. Create a list of likely expenses. This list must include rent/mortgage, utilities, food, and school loan repayment. You may also want to include car payments, gasoline, and other items.
5. Categorize each expense as need, want, or savings.
6. Using the amounts found in step 3, decide how much to allocate to each of your expenses. It may help to quickly research how much rent is where you want to live.
7. Discuss the choices you had to make, and why you prioritized some expenses over others.

### Interest Rate and Time: What Is the Relationship?

The interplay between interest rate and time for an annuity is not easily seen. How the amount that must be deposited per compounding period, , changes based on the time and interest rate would be useful to understand. In this project, you will explore this relationship. We will use a fixed future value of  = $1,000,000 and a fixed number of periods per year, 12 (monthly compounding). With those, we’ll find various annuity payments that must be made to reach the goal.
The annual interest rate for the investment is in the top row. The number of years for the investment is in the left column. In each cell (or box), find the monthly payment necessary to reach the goal of $1,000,000.
Describe how the interest rates and number of years impact the payment necessary to reach the goal of $1,000,000.

### Finding a Home

In this project you will identify a home you like, and then estimate the costs associated with that home.
1. Find a home in your region that you would like to buy using an online search of listings in your area. Zillow is a good place to begin.
2. Find the asking price for this home. Assume you would pay that price.
3. Find an estimation for closing costs in your area. Assume you finance those costs also.
4. Estimate the taxes to be paid on the home per year. It is likely that the online listing of the home has an estimate for the taxes for the house.
5. Use Google to determine the average homeowner’s insurance cost in your region.
6. Use the internet to determine the average interest rate for a 30-year mortgage.
7. Find how much you would pay per month, based on the answers to the previous questions, including the escrow payments for taxes and insurance.
8. Assume you will pay $50 per $100,000 borrowed in PMI. Add this to the monthly payment.

### Chapter Review

### Understanding Percent

### Discounts, Markups, and Sales Tax

### Simple Interest

### Compound Interest

### Making a Personal Budget

### Methods of Savings

### Investments

### The Basics of Loans

### Understanding Student Loans

### Credit Cards

### Buying or Leasing a Car

### Renting and Homeownership

### Income Tax

### Chapter Test


# Probability

## Introduction

Casinos are big business; according to the American Gaming Association, commercial casinos in the United States brought in over $43 billion in revenue in 2019. Casinos must walk a fine line in order to be profitable. Their customers must lose more money than they win, on average, in order to stay in business. But if the chances of a single customer winning more money than they lose is too small, people will stop coming in the door to play the games.
In this chapter, we'll study the techniques a casino must use to determine how likely it is that a customer will win a particular game, and then how the casino decides how much money a winner will rake in so that the customers are happy, but the casino also turns a profit in the long run. In order to figure out those likelihoods, we have to be able to somehow consider every possible outcome of these games. For example, in a game that involves players receiving 5 cards from a deck of 52, there are 2,598,960 possibilities for each player. We'll start off this chapter by learning how to count those possible outcomes.


# Probability

## The Multiplication Rule for Counting

### Learning Objectives

After completing this section, you should be able to:
1. Apply the Multiplication Rule for Counting to solve problems.
One of the first bits of mathematical knowledge children learn is how to count objects by pointing to them in turn and saying: “one, two, three, …” That’s a useful skill, but when the number of things that we need to count grows large, that method becomes onerous (or, for very large numbers, impossible for humans to accomplish in a typical human lifespan). So, mathematicians have developed short cuts to counting big numbers. These techniques fall under the mathematical discipline of combinatorics, which is devoted to counting.

### Multiplication as a Combinatorial Short Cut

One of the first combinatorial short cuts to counting students learn in school has to do with areas of rectangles. If we have a set of objects to be counted that can be physically arranged into a rectangular shape, then we can use multiplication to do the counting for us. Consider this set of objects ():
Certainly we can count them by pointing and running through the numbers, but it’s more efficient to group them ().

If we group the balls by 4s, we see that we have 6 groups (or, we can see this arrangement as 4 groups of 6 balls). Since multiplication is repeated addition (i.e., ), we can use this grouping to quickly see that there are 24 balls.
Let’s generalize this idea a little bit. Let’s say that we’re visiting a bakery that offers customized cupcakes. For the cake, we have three choices: vanilla, chocolate, and strawberry. Each cupcake can be topped with one of four types of frosting: vanilla, chocolate, lemon, and strawberry. How many different cupcake combinations are possible? We can think of laying out all the possibilities in a grid, with cake choices defining the rows and frosting choices defining the columns (Figure 7.5).
Since there are 3 rows (cakes) and 4 columns (frostings), we have  possible combinations. This is the reasoning behind the Multiplication Rule for Counting, which is also known as the Fundamental Counting Principle. This rule says that if there are  ways to accomplish one task and  ways to accomplish a second task, then there are  ways to accomplish both tasks. We can tack on additional tasks by multiplying the number of ways to accomplish those tasks to our previous product.

### Check Your Understanding

### Key Terms

1. combinatorics
2. Multiplication Rule for Counting (Fundamental Counting Principle)

### Key Concepts

1. The Multiplication Rule for Counting is used to count large sets.


# Probability

## Permutations

### Learning Objectives

After completing this section, you should be able to:
1. Use the Multiplication Rule for Counting to determine the number of permutations.
2. Compute expressions containing factorials.
3. Compute permutations.
4. Apply permutations to solve problems.
Swimming events are some of the most popular events at the summer Olympic Games. In the finals of each event, 8 swimmers compete at the same time, making for some exciting finishes. How many different orders of finish are possible in these events? In this section, we’ll extend the Multiplication Rule for Counting to help answer questions like this one, which relate to permutations. A permutation is an ordered list of objects taken from a given population. The length of the list is given, and the list cannot contain any repeated items.

### Applying the Multiplication Rule for Counting to Permutations

In the case of the swimming finals, one possible permutation of length 3 would be the list of medal winners (first, second, and third place finishers). A permutation of length 8 would be the full order of finish (first place through eighth place). Let’s use the Multiplication Rule for Counting to figure out how many of each of these permutations there are.

### Factorials

The pattern we see in  occurs commonly enough that we have a name for it: factorial. For any positive whole number , we define the factorial of  (denoted  and read " factorial") to be the product of every whole number less than or equal to . We also define 0! to be equal to one. We will use factorials in a couple of different contexts, so let's get some practice doing computations with them.

### Permutations

As we’ve seen, factorials can pop up when we’re computing permutations. In fact, there is a formula that we can use to make that connection explicit. Let’s define some notation first. If we have a collection of  objects and we wish to create an ordered list of  of the objects (where ), we’ll call the number of those permutations  (read “the number of permutations of  objects taken  at a time”). We formalize the formula we'll use to compute permutations below.
If you wondered why we defined  earlier, it was to make formulas like this one work; if we have  objects and want to order all of them (so, we want the number of permutations of  objects taken  at a time), we get . Next, we’ll get some practice computing these permutations.

### Check Your Understanding

### Key Terms

1. permutation
2. factorial

### Key Concepts

1. Using the Multiplication Rule for Counting to enumerate permutations.
2. Simplifying and computing expressions involving factorials.
3. Using factorials to count permutations.

### Formulas

1.


# Probability

## Combinations

### Learning Objectives

After completing this section, you should be able to:
1. Distinguish between permutation and combination uses.
2. Compute combinations.
3. Apply combinations to solve applications.
In Permutations, we studied permutations, which we use to count the number of ways to generate an ordered list of a given length from a group of objects. An important property of permutations is that the order of the list matters: The results of a race and the selection of club officers are examples of lists where the order is important. In other situations, the order is not important. For example, in most card games where a player receives a hand of cards, the order in which the cards are received is irrelevant; in fact, players often rearrange the cards in a way that helps them keep the cards organized.

### Combinations: When Order Doesn’t Matter

In situations in which the order of a list of objects doesn’t matter, the lists are no longer permutations. Instead, we call them combinations.

### Counting Combinations

Permutations and combinations are certainly related, because they both involve choosing a subset of a large group. Let’s explore that connection, so that we can figure out how to use what we know about permutations to help us count combinations. We’ll take a basic example. How many ways can we select 3 letters from the group A, B, C, D, and E? If order matters, that number is . That’s small enough that we can list them all out in the table below.
Now, let’s look back at that list and color-code it so that groupings of the same 3 letters get the same color, as shown in :
After color-coding, we see that the 60 cells can be seen as 10 groups (colors) of 6. That’s no coincidence! We’ve already seen how to compute the number of permutations using the formula To compute the number of combinations, let’s count them another way using the Multiplication Rule for Counting. We’ll do this in two steps:
Step 1: Choose 3 letters (paying no attention to order).
Step 2: Put those letters in order.
The number of ways to choose 3 letters from this group of 5 (A, B, C, D, E) is the number of combinations we’re looking for; let’s call that number  (read “the number of combinations of 5 objects taken 3 at a time”). We can see from our chart that this is ten (the number of colors used). We can generalize our findings this way: remember that the number of permutations of  things taken  at a time is . That number is also equal to , and so it must be the case that . Dividing both sides of that equation by  gives us the formula below.

### Check Your Understanding

### Key Terms

1. combination

### Key Concepts

1. Permutations are used to count subsets when order matters; combinations work when order doesn't matter.
2. Combinations can also be computed using factorials.

### Formulas

1. The formula for counting combinations is:


# Probability

## Tree Diagrams, Tables, and Outcomes

### Learning Objectives

After completing this section, you should be able to:
1. Determine the sample space of single stage experiment.
2. Use tables to list possible outcomes of a multistage experiment.
3. Use tree diagrams to list possible outcomes of a multistage experiment.
In the 19th century, an Augustinian friar and scientist named Gregor Mendel used his observations of pea plants to set out his theory of genetic propagation. In his work, he looked at the offspring that resulted from breeding plants with different characteristics together. For applications like this, it is often insufficient to only know in how many ways a process might end; we need to be able to list all of the possibilities. As we’ve seen, the number of possible outcomes can be very large! Thus, it’s important to have a strategy that allows us to systematically list these possibilities to make sure we don’t leave any out. In this section, we’ll look at two of these strategies.

### Single Stage Experiments

When we are talking about combinatorics or probability, the word “experiment” has a slightly different meaning than it does in the sciences. Experiments can range from very simple (“flip a coin”) to very complex (“count the number of uranium atoms that undergo nuclear fission in a sample of a given size over the course of an hour”). Experiments have unknown outcomes that generally rely on something random, so that if the experiment is repeated (or replicated) the outcome might be different. No matter what the experiment, though, analysis of the experiment typically begins with identifying its sample space.
The sample space of an experiment is the set of all of the possible outcomes of the experiment, so it’s often expressed as a set (i.e., as a list bound by braces; if the experiment is “randomly select a number between 1 and 4,” the sample space would be written ).

### Multistage Experiments

Some experiments have more complicated sample spaces because they occur in stages. These stages can occur in succession (like drawing cards one at a time) or simultaneously (rolling 2 dice). Sample spaces get more complicated as the complexity of the experiment increases, so it’s important to choose a systematic method for identifying all of the possible outcomes. The first method we’ll discuss is the table.

### Using Tables to Find Sample Spaces

Tables are useful for finding the sample space for experiments that meet two criteria: (1) The experiment must have only two stages, and (2) the outcomes of each stage must have no effect on the outcomes of the other. When the stages do not affect each other, we say the stages are independent. Otherwise, the stages are dependent and so we can’t use tables; we’ll look at a method for analyzing dependent stages soon.
If you have a two-stage experiment with independent stages, a table is the most straightforward way to identify the sample space. To build a table, you list the outcomes of one stage of the experiment along the top of the table and the outcomes of the other stage down the side. The cells in the interior of the table are then filled using the outcomes associated with each cell’s row and column. Let’s look at an example.

### Using Tree Diagrams to Identify Sample Spaces

In experiments where there are more than two stages, or where the stages are dependent, a tree diagram is a helpful tool for systematically identifying the sample space. Tree diagrams are built by first drawing a single point (or node), then from that node we draw one branch (a short line segment) for each outcome of the first stage. Each branch gets its own node at the other end (which we typically label with the corresponding outcome for that branch); from each of these, we draw another branch for each outcome of the second stage, assuming that the outcome of the first stage matches the branch we were on. If there are other stages, we can continue from there by continuing to add branches and nodes. This sounds really complicated, but it’s easier to understand through an example.

### Check Your Understanding

### Key Terms

1. experiment
2. replication
3. sample space
4. independent/dependent

### Key Concepts

1.  We identify the sample space of an experiment by identifying all of its possible outcomes.
2.  Tables can help us find a sample space by keeping the possible outcomes organized.
3.  Tree diagrams provide a visualization of the sample space of an experiment that involves multiple stages.


# Probability

## Basic Concepts of Probability

### Learning Objectives

After completing this section, you should be able to:
1. Define probability including impossible and certain events.
2. Calculate basic theoretical probabilities.
3. Calculate basic empirical probabilities.
4. Distinguish among theoretical, empirical, and subjective probability.
5. Calculate the probability of the complement of an event.
It all comes down to this. The game of Monopoly that started hours ago is in the home stretch. Your sister has the dice, and if she rolls a 4, 5, or 7 she’ll land on one of your best spaces and the game will be over. How likely is it that the game will end on the next turn? Is it more likely than not? How can we measure that likelihood? This section addresses this question by introducing a way to measure uncertainty.

### Introducing Probability

Uncertainty is, almost by definition, a nebulous concept. In order to put enough constraints on it that we can mathematically study it, we will focus on uncertainty strictly in the context of experiments. Recall that experiments are processes whose outcomes are unknown; the sample space for the experiment is the collection of all those possible outcomes. When we want to talk about the likelihood of particular outcomes, we sometimes group outcomes together; for example, in the Monopoly example at the beginning of this section, we were interested in the roll of 2 dice that might fall as a 4, 5, or 7. A grouping of outcomes that we’re interested in is called an event. In other words, an event is a subset of the sample space of an experiment; it often consists of the outcomes of interest to the experimenter.
Once we have defined the event that interests us, we can try to assess the likelihood of that event. We do that by assigning a number to each event () called the probability of that event (). The probability of an event is a number between 0 and 1 (inclusive). If the probability of an event is 0, then the event is impossible. On the other hand, an event with probability 1 is certain to occur. In general, the higher the probability of an event, the more likely it is that the event will occur.

### Three Ways to Assign Probabilities

The probabilities of events that are certain or impossible are easy to assign; they’re just 1 or 0, respectively. What do we do about those in-between cases, for events that might or might not occur? There are three methods to assign probabilities that we can choose from. We’ll discuss them here, in order of reliability.

### Method 1: Theoretical Probability

The theoretical method gives the most reliable results, but it cannot always be used. If the sample space of an experiment consists of equally likely outcomes, then the theoretical probability of an event is defined to be the ratio of the number of outcomes in the event to the number of outcomes in the sample space.

### Method 2: Empirical Probability

Theoretical probabilities are precise, but they can’t be found in every situation. If the outcomes in the sample space are not equally likely, then we’re out of luck. Suppose you’re watching a baseball game, and your favorite player is about to step up to the plate. What is the probability that he will get a hit?
In this case, the sample space is {hit, not a hit}. That doesn’t mean that the probability of a hit is , since those outcomes aren’t equally likely. The theoretical method simply can’t be used in this situation. Instead, we might look at the player’s statistics up to this point in the season, and see that he has 122 hits in 531 opportunities. So, we might think that the probability of a hit in the next plate appearance would be about . When we use the outcomes of previous replications of an experiment to assign a probability to the next replication, we’re defining an empirical probability. Empirical probability is assigned using the outcomes of previous replications of an experiment by finding the ratio of the number of times in the previous replications the event occurred to the total number of previous replications.
Empirical probabilities aren’t exact, but when the number of previous replications is large, we expect them to be close. Also, if the previous runs of the experiment are not conducted under the exact set of circumstances as the one we’re interested in, the empirical probability is less reliable. For instance, in the case of our favorite baseball player, we might try to get a better estimate of the probability of a hit by looking only at his history against left- or right-handed pitchers (depending on the handedness of the pitcher he’s about to face).

### Method 3: Subjective Probability

In cases where theoretical probability can’t be used and we don’t have prior experience to inform an empirical probability, we’re left with one option: using our instincts to guess at a subjective probability. A subjective probability is an assignment of a probability to an event using only one’s instincts.
Subjective probabilities are used in cases where an experiment can only be run once, or it hasn’t been run before. Because subjective probabilities may vary widely from person to person and they’re not based on any mathematical theory, we won’t give any examples. However, it’s important that we be able to identify a subjective probability when we see it; they will in general be far less accurate than empirical or theoretical probabilities.

### New Probabilities from Old: Complements

One of the goals of the rest of this chapter is learning how to break down complicated probability calculations into easier probability calculations. We’ll look at the first of the tools we can use to accomplish this goal in this section; the rest will come later.
Given an event , the complement of  (denoted ) is the collection of all of the outcomes that are not in . (This is language that is taken from set theory, which you can learn more about elsewhere in this text.) Since every outcome in the sample space either is or is not in , it follows that . So, if the outcomes in  are equally likely, we can compute theoretical probabilities  and . Then, adding these last two equations, we get

Thus, if we subtract  from both sides, we can conclude that . Though we performed this calculation under the assumption that the outcomes in  are all equally likely, the last equation is true in every situation.
How is this helpful? Sometimes it is easier to compute the probability that an event won’t happen than it is to compute the probability that it will. To apply this principle, it’s helpful to review some tricks for dealing with inequalities. If an event is defined in terms of an inequality, the complement will be defined in terms of the opposite inequality: Both the direction and the inclusivity will be reversed, as shown in the table below.

### Check Your Understanding

### Key Terms

1. event
2. probability
3. theoretical probability
4. empirical probability
5. subjective probability

### Key Concepts

1. The theoretical probability of an event is the ratio of the number of equally likely outcomes in the event to the number of equally likely outcomes in the sample space.
2. Empirical probabilities are computed by repeating the experiment many times, and then dividing the number of replications that result in the event of interest by the total number of replications.
3. Subjective probabilities are assigned based on subjective criteria, usually because the experiment can’t be repeated and the outcomes in the sample space are not equally likely.
4. The probability of the complement of an event is found by subtracting the probability of the event from one.

### Formulas

1. For an experiment whose sample space  consists of equally likely outcomes, the theoretical probability of the event  is the ratio  where  and  denote the number of outcomes in the event and in the sample space, respectively.
2.


# Probability

## Probability with Permutations and Combinations

### Learning Objectives

After completing this section, you should be able to:
1. Calculate probabilities with permutations.
2. Calculate probabilities with combinations.
In our earlier discussion of theoretical probabilities, the first step we took was to write out the sample space for the experiment in question. For many experiments, that method just isn’t practical. For example, we might want to find the probability of drawing a particular 5-card poker hand. Since there are 52 cards in a deck and the order of cards doesn’t matter, the sample space for this experiment has  possible 5-card hands. Even if we had the patience and space to write them all out, sorting through the results to find the outcomes that fall in our event would be just as tedious.
Luckily, the formula for theoretical probabilities doesn’t require us to know every outcome in the sample space; we just need to know how many outcomes there are. In this section, we’ll apply the techniques we learned earlier in the chapter (The Multiplication Rule for Counting, permutations, and combinations) to compute probabilities.

### Using Permutations to Compute Probabilities

Recall that we can use permutations to count how many ways there are to put a number of items from a list in order. If we’re looking at an experiment whose sample space looks like an ordered list, then permutations can help us to find the right probabilities.

### Combinations to Computer Probabilities

If the sample space of our experiment is one in which order doesn’t matter, then we can use combinations to find the number of outcomes in that sample space.

### Check Your Understanding

### Key Concepts

1. We use permutations and combinations to count the number of equally likely outcomes in an event and in a sample space, which allows us to compute theoretical probabilities.


# Probability

## What Are the Odds?

### Learning Objectives

After completing this section, you should be able to:
1. Compute odds.
2. Determine odds from probabilities.
3. Determine probabilities from odds.
A particular lottery instant-win game has 2 million tickets available. Of those, 500,000 win a prize. If there are 500,000 winners, then it follows that there are 1,500,000 losing tickets. When we evaluate the risk associated with a game like this, it can be useful to compare the number of ways to win the game to the number of ways to lose. In the case of this game, we would compare the 500,000 wins to the 1,500,000 losses. In other words, there are 3 losing tickets for every winning ticket. Comparisons of this type are the focus of this section.

### Computing Odds

The ratio of the number of equally likely outcomes in an event  to the number of equally likely outcomes not in the event  is called the odds for (or odds in favor of) the event. The opposite ratio (the number of outcomes not in the event to the number in the event  to the number in the event  is called the odds against the event.

### Odds as a Ratio of Probabilities

We can also think of odds as a ratio of probabilities. Consider again the instant-win game from the section opener, with 500,000 winning tickets out of 2,000,000 total tickets. If a player buys one ticket, the probability of winning is , and the probability of losing is . Notice that the ratio of the probability of winning to the probability of losing is , which matches the odds in favor of winning.
We can use these formulas to convert probabilities to odds, and vice versa.
Now, let’s convert odds to probabilities. Let’s say the odds for an event are . Then, using the formula above, we have . Converting to fractions and solving for , we get:
Let’s put this result in a formula we can use.

### Check Your Understanding

### Key Terms

1. odds (for/against)

### Key Concepts

1. Odds are computed as the ratio of the probability of an event to the probability of its compliment.

### Formulas

1. For an event ,

2. If the odds in favor of  are , then .


# Probability

## The Addition Rule for Probability

### Learning Objectives

After completing this section, you should be able to:
1. Identify mutually exclusive events.
2. Apply the Addition Rule to compute probability.
3. Use the Inclusion/Exclusion Principle to compute probability.
Up to this point, we have looked at the probabilities of simple events. Simple events are those with a single, simple characterization. Sometimes, though, we want to investigate more complicated situations. For example, if we are choosing a college student at random, we might want to find the probability that the chosen student is a varsity athlete or in a Greek organization. This is a compound event: there are two possible criteria that might be met. We might instead try to identify the probability that the chosen student is both a varsity athlete and in a Greek organization. In this section and the next, we’ll cover probabilities of two types of compound events: those build using “or” and those built using “and.” We’ll deal with the former first.

### Mutual Exclusivity

Before we get to the key techniques of this section, we must first introduce some new terminology. Let’s say you’re drawing a card from a standard deck. We’ll consider 3 events:  is the event “the card is a ,”  is the event “the card is a 10,” and  is the event “the card is a .” If the card drawn is , then  and  didn’t occur, but  did. If the card drawn is instead , then  didn’t occur, but both  and  did.
We can see from these examples that, if we are interested in several possible events, more than one of them can occur simultaneously (both  and , for example). But, if you think about all the possible outcomes, you can see that  and  can never occur simultaneously; there are no cards in the deck that are both  and . Pairs of events that cannot both occur simultaneously are called mutually exclusive. Let’s go through an example to help us better understand this concept.

### The Addition Rule for Mutually Exclusive Events

If two events are mutually exclusive, then we can use addition to find the probability that one or the other event occurs.
Why does this formula work? Let’s consider a basic example. Suppose we’re about to draw a Scrabble tile from a bag containing A, A, B, E, E, E, R, S, S, U. What is the probability of drawing an E or an S? Since 3 of the tiles are marked with E and 2 are marked with S, there are 5 tiles that satisfy the criteria. There are ten tiles in the bag, so the probability is . Notice that the probability of drawing an E is  and the probability of drawing an S is ; adding those together, we get . Look at the numerators in the fractions involved in the sum: the 3 represents the number of E tiles and the 2 is the number of S tiles. This is why the Addition Rule works: The total number of outcomes in one event or the other is the sum of the numbers of outcomes in each of the individual events.

### Finding Probabilities When Events Aren’t Mutually Exclusive

Let’s return to the example we used to explore the Addition Rule: We’re about to draw a Scrabble tile from a bag containing A, A, B, E, E, E, R, S, S, U. Consider these events:  is “draw a vowel” and  is “draw a letter that comes after L in the alphabet.” Since there are 6 vowels, . There are 4 tiles with letters that come after L alphabetically, so . What is ? If we blindly apply the Addition Rule, we get , which would mean that the compound event  or  is certain. However, it’s possible to draw a B, in which case neither  nor  happens. Where’s the error?
The events are not mutually exclusive: the outcome U belongs to both events, and so the Addition Rule doesn’t apply. However, there’s a way to extend the Addition Rule to allow us to find this probability anyway; it’s called the Inclusion/Exclusion Principle. In this example, if we just add the two probabilities together, the outcome U is included in the sum twice: It’s one of the 6 outcomes represented in the numerator of , and it’s one of the 4 outcomes represented in the numerator of . So, that particular outcome has been “double counted.” Since it has been included twice, we can get a true accounting by excluding it once: . We can generalize this idea to a formula that we can apply to find the probability of any compound event built using “or.”
It’s worth noting that this formula is truly an extension of the Addition Rule. Remember that the Addition Rule requires that the events  and  are mutually exclusive. In that case, the compound event  is impossible, and so . So, in cases where the events in question are mutually exclusive, the Inclusion/Exclusion Principle reduces to the Addition Rule.

### Check Your Understanding

### Key Terms

1. mutually exclusive

### Key Concepts

1. The Addition Rule is used to find the probability that one event or another will occur when those events are mutually exclusive.
2. The Inclusion/Exclusion Principle is used to find probabilities when events are not mutually exclusive.

### Formulas

1. If  and  are mutually exclusive events, then

2. If  and  are events that contain outcomes of a single experiment, then


# Probability

## Conditional Probability and the Multiplication Rule

### Learning Objectives

After completing this section, you should be able to:
1. Calculate conditional probabilities.
2. Apply the Multiplication Rule for Probability to compute probabilities.
Back in , we constructed the following table () to help us find the probabilities associated with rolling two standard 6-sided dice:
For example, 3 of these 36 equally likely outcomes correspond to rolling a sum of 10, so the probability of rolling a 10 is . However, if you choose to roll the dice one at a time, the probability of rolling a 10 will change after the first die comes to rest. For example, if the first die shows a 5, then the probability of rolling a sum of 10 has jumped to —the event will occur if the second die also shows a 5, which is 1 of 6 equally likely outcomes for the second die. If instead the first die shows a 3, then the probability of rolling a sum of 10 drops to 0—there are no outcomes for the second die that will give us a sum of 10.
Understanding how probabilities can shift as we learn new information is critical in the analysis of our second type of compound events: those built with “and.” This section will explain how to compute probabilities of those compound events.

### Conditional Probabilities

When we analyze experiments with multiple stages, we often update the probabilities of the possible final outcomes or the later stages of the experiment based on the results of one or more of the initial stages. These updated probabilities are called conditional probabilities.
In other words, if  is a possible outcome of the first stage in a multistage experiment, then the probability of an event  conditional on  (denoted , read “the probability of  given ”) is the updated probability of  under the assumption that  occurred.
In the example that opened this section, we might consider rolling two dice as a multistage experiment: rolling one, then the other. If we define  to be the event “roll a sum of 10,”  to be the event “first die shows 5,” and  to be the event “first die shows 3,” then we computed , , and .

### Compound Events Using “And” and the Multiplication Rule

For multistage experiments, the outcomes of the experiment as a whole are often stated in terms of the outcomes of the individual stages. Commonly, those statements are joined with “and.” For example, in the sock drawer example just above, one outcome might be “the left sock is black and the right sock is blue.” As with “or” compound events, these probabilities can be computed with basic arithmetic.
It is often useful to combine the rules we’ve seen so far with the techniques we used for finding sample spaces. In particular, trees can be helpful when we want to identify the probabilities of every possible outcome in a multistage experiment. The next example will illustrate this.

### Check Your Understanding

### Key Terms

1. conditional probability

### Key Concepts

1. Conditional probabilities are computed under the assumption that the condition has already occurred.
2. The Multiplication Rule for Probability is used to find the probability that two events occur in sequence.

### Formulas

1. If  and  are events associated with the first and second stages of an experiment, then .


# Probability

## The Binomial Distribution

### Learning Objectives

After completing this section, you should be able to:
1. Identify binomial experiments.
2. Use the binomial distribution to analyze binomial experiments.
It’s time for the World Series, which determines the champion for this season in Major League Baseball. The scrappy Los Angeles Angels are facing the powerhouse Cincinnati Reds. Computer models put the chances of the Reds winning any single game against the Angels at about 65%. The World Series, as its name implies, isn’t just one game, though: it’s what’s known as a “best-of-seven” contest: the teams play each other repeatedly until one team wins 4 games (which could take up to 7 games total, if each team wins three of the first 6 matchups). If the Reds truly have a 65% chance of winning a single game, then the probability that they win the series should be greater than 65%. Exactly how much bigger?
If you have the patience for it, you could use a tree diagram like we used in  to trace out all of the possible outcomes, find all the related probabilities, and add up the ones that result in the Reds winning the series. Such a tree diagram would have  final nodes, though, so the calculations would be very tedious. Fortunately, we have tools at our disposal that allow us to find these probabilities very quickly. This section will introduce those tools and explain their use.

### Binomial Experiments

The tools of this section apply to multistage experiments that satisfy some pretty specific criteria. Before we move on to the analysis, we need to introduce and explain those criteria so that we can recognize experiments that fall into this category. Experiments that satisfy each of these criteria are called binomial experiments. A binomial experiment is an experiment with a fixed number of repeated independent binomial trials, where each trial has the same probability of success.

### Repeated Binomial Trials

The first criterion involves the structure of the stages. Each stage of the experiment should be a replication of every other stage; we call these replications trials. An example of this is flipping a coin 10 times; each of the ten flips is a trial, and they all occur under the same conditions as every other. Further, each trial must have only two possible outcomes. These two outcomes are typically labeled “success” and “failure,” even if there is not a positive or negative connotation associated with those outcomes. Experiments with more than two outcomes in their sample spaces are sometimes reconsidered in a way that forces just two outcomes; all we need to do is completely divide the sample space into two parts that we can label “success” and “failure.” For example, your grade on an exam might be recorded as A, B, C, D, or F, but we could instead think of the grades A, B, C, and D as “success” and a grade of F as “failure.” Trials with only two outcomes are called binomial trials (the word binomial derives from Latin and Greek roots that mean “two parts”).

### Independent Trials

The next criterion that we’ll be looking for is independence of trials. Back in Tree Diagrams, Tables, and Outcomes, we said that two stages of an experiment are independent if the outcome of one stage doesn’t affect the other stage. Independence is necessary for the experiments we want to analyze in this section.

### Fixed Number of Trials

Next, we require that the number of trials in the experiment be decided before the experiment begins. For example, we might say “flip a coin 10 times.” The number of trials there is fixed at 10. However, if we say “flip a coin until you get 5 heads,” then the number of trials could be as low as 5, but theoretically it could be 50 or a 100 (or more)! We can’t apply the tools from this section in cases where the number of trials is indeterminate.

### Constant Probability

The next criterion needed for binomial experiments is related to the independence of the trials. We must make sure that the probability of success in each trial is the same as the probability of success in every other trial.

### The Binomial Formula

If we flip a coin 100 times, you might expect the number of heads to be around 50, but you probably wouldn’t be surprised if the actual number of heads was 47 or 52. What is the probability that the number of heads is exactly 50? Or falls between 45 and 55? It seems unlikely that we would get more than 70 heads. Exactly how unlikely is that?
Each of these questions is a question about the number of successes in a binomial experiment (flip a coin 100 times, “success” is flipping heads). We could theoretically use the techniques we’ve seen in earlier sections to answer each of these, but the number of calculations we’d have to do is astronomical; just building the tree diagram that represents this situation is more than we could complete in a lifetime; it would have  final nodes! To put that number in perspective, if we could draw 1,000 dots every second, and we started at the moment of the Big Bang, we’d currently be about 0.00000003% of the way to drawing out those final nodes. Luckily, there’s a shortcut called the Binomial Formula that allows us to get around doing all those calculations!
We can use this formula to answer one of our questions about 100 coin flips. What is the probability of flipping exactly 50 heads? In this case, , , and , so . Unfortunately, many calculators will balk at this calculation; that first factor () is an enormous number, and the other two factors are very close to zero. Even if your calculator can handle numbers that large or small, the arithmetic can create serious errors in rounding off.

### The Binomial Distribution

If we are interested in the probability of more than just a single outcome in a binomial experiment, it’s helpful to think of the Binomial Formula as a function, whose input is the number of successes and whose output is the probability of observing that many successes. Generally, for a small number of trials, we’ll give that function in table form, with a complete list of the possible outcomes in one column and the probability in the other.
For example, suppose Kristen is practicing her basketball free throws. Assume Kristen always makes 82% of those shots. If she attempts 5 free throws, then the Binomial Formula gives us these probabilities:
A table that lists all possible outcomes of an experiment along with the probabilities of those outcomes is an example of a probability density function (PDF). A PDF may also be a formula that you can use to find the probability of any outcome of an experiment.
If we want to know the probability of a range of outcomes, we could add up the corresponding probabilities. Going back to Kristen’s free throws, we can find the probability that she makes 3 or fewer of her 5 attempts by adding up the probabilities associated with the corresponding outcomes (in this case: 0, 1, 2, or 3):
The probability that the outcome of an experiment is less than or equal to a given number is called a cumulative probability. A table of the cumulative probabilities of all possible outcomes of an experiment is an example of a cumulative distribution function (CDF). A CDF may also be a formula that you can use to find those cumulative probabilities.
Here are the PDF and CDF for Kristen’s free throws:
Finally, we can answer the question posed at the beginning of this section. Remember that the Reds are facing the Angels in the World Series, which is won by the team who is first to win 4 games. The Reds have a 65% chance to win any game against the Angels. So, what is the probability that the Reds win the World Series? At first glance, this is not a binomial experiment: The number of games played is not fixed, since the series ends as soon as one team wins 4 games. However, we can extend this situation to a binomial experiment: Let’s assume that 7 games are always played in the World Series, and the winner is the team who wins more games. In a way, this is what happens in reality; it’s as though the first team to lose 4 games (and thus cannot win more than the other team) forfeits the rest of their games. So, we can treat the actual World Series as a binomial experiment with seven trials. If  is the number of games won by the Reds, the probability that the Reds win the World Series is . Using the techniques from the last example, we get .

### Check Your Understanding

### Key Terms

1. binomial experiment
2. probability density function (PDF)
3. cumulative distribution function (CDF)

### Key Concepts

1. Binomial experiments result when we count the number of successful outcomes in a fixed number of repeated, independent trials with a constant probability of success.
2. The binomial distribution is used to find probabilities associated with binomial experiments.
3. Probability density functions (PDFs) describe the probabilities of individual outcomes in an experiment; cumulative distribution functions (CDFs) give the probabilities of ranges of outcomes.

### Formulas

1. Suppose we have a binomial experiment with  trials and the probability of success in each trial is . Then:


# Probability

## Expected Value

### Learning Objectives

After completing this section, you should be able to:
1. Calculate the expected value of an experiment.
2. Interpret the expected value of an experiment.
3. Use expected value to analyze applications.
The casino game roulette has dozens of different bets that can be made. These bets have different probabilities of winning but also have different payouts. In general, the lower the probability of winning a bet is, the more money a player wins for that bet. With so many options, is there one bet that’s “smarter” than the rest? What’s the best play to make at a roulette table? In this section, we’ll develop the tools we need to answer these questions.

### Expected Value

Many experiments have numbers associated with their outcomes. Some are easy to define; if you roll 2 dice, the sum of the numbers showing is a good example. In some card games, cards have different point values associated with them; for example, in some forms of the game rummy, aces are worth 15 points; 10s, jacks, queens, and kings are worth 10; and all other cards are worth 5. The outcomes of casino and lottery games are all associated with an amount of money won or lost. These outcome values are used to find the expected value of an experiment: the mean of the values associated with the outcomes that we would observe over a large number of repetitions of the experiment. (See Conditional Probability and the Multiplication Rule for more on means.)
That definition is a little vague; How many is “a large number?” In practice, it depends on the experiment; the number has to be large enough that every outcome would be expected to appear at least a few times. For example, if we’re talking about rolling a standard 6-sided die and we note the number showing, a few dozen replications should be enough that the mean would be representative. Since the probability of each outcome is , we would expect to see each outcome about 8 times over the course of 48 replications. However, if we’re talking about the Powerball lottery, where the probability of winning the jackpot is about , we would need several billion replications to ensure that every outcome appears a few times. Luckily, we can find the theoretical expected value before we even run the experiment the first time.
Let’s make note of some things we can learn from . First, as Exercises 1 and 3 demonstrate, the expected value of an experiment might not be a value that could come up in the experiment. Remember that the expected value is interpreted as a mean, and the mean of a collection of numbers doesn’t have to actually be one of those numbers.
 Second, looking at Exercise 1, the expected value (3.5) was just the mean of the numbers on the faces of the die: . This is no accident! If we break that fraction up using the addition in the numerator, we get , which we can rewrite as . That’s exactly the computation we did to find the expected value! In fact, expected values can always be treated as a special kind of mean called a weighted mean, where the weights are the probabilities associated with each value. When the probabilities are all equal, the weighted mean is just the regular mean.

### Interpreting Expected Values

As we noted, the expected value of an experiment is the mean of the values we would observe if we repeated the experiment a large number of times. (This interpretation is due to an important theorem in the theory of probability called the Law of Large Numbers.) Let’s use that to interpret the results of the previous example.

### Using Expected Value

Now that we know how to find and interpret expected values, we can turn our attention to using them. Suppose someone offers to play a game with you. If you roll a die and get a 6, you get $10. However, if you get a 5 or below, you lose $1. Is this a game you’d want to play? Let’s look at the expected value: The probability of winning is  and the probability of losing is , so the expected value is . That means, on average, you’ll come out ahead by about 83 cents every time you play this game. It’s a great deal! On the other hand, if the winnings for rolling a 6 drop to $3, the expected value becomes , meaning you should expect to lose about 33 cents on average for every time you play. Playing that game is not a good idea! In general, this is how casinos and lottery corporations make money: Every game has a negative expected value for the player.

### Check Your Understanding

### Key Terms

1. expected value

### Key Concepts

1. The expected value of an experiment is the sum of the products of the numerical outcomes of an experiment with their corresponding probabilities.
2. The expected value of an experiment is the most likely value of the average of a large number of replications of the experiment.

### Formulas

1. If  represents an outcome of an experiment and  represents the value of that outcome, then the expected value of the experiment is:

    where

### Projects

1. The Binomial Distribution is one of many examples of a discrete probability distribution. Other examples include the Geometric, Hypergeometric, Multinomial, Poisson, and Negative Binomial Distributions. Choose one of these distributions, and find out what makes it different from the Binomial Distribution. In what situations can it be applied? How is it used? Once you have an idea of how it’s used, write a series of five questions like the ones in this chapter that can be answered with that distribution, and find the answers.
2. Binomial is a word that also comes up in algebra; the word describes polynomials with two terms. At first glance, there isn’t much to indicate that these two uses of the word are related, but it turns out there is a connection. Explore the connection between the Binomial Distribution and the algebraic concept of binomial expansion, (the process of multiplying out expressions like  for a positive whole number ). Search for a connection with the mathematical object known as Pascal’s Triangle.
3. Hazard is a dice game that was mentioned in Chaucer’s Canterbury Tales. It was a popular game of chance played in taverns and coffee houses well into the 18th century; its popularity at the time of the foundation of probability theory means that it was a common example in early texts on finding expected values and probabilities. Find the rules of the game, and get some practice playing it. Then, analyze the choices that the caster gets to make, and decide which is most advantageous, using the language of expected values.

### Chapter Review

### The Multiplication Rule for Counting

### Permutations

### Combinations

### Tree Diagrams, Tables, and Outcomes

### Basic Concepts of Probability

### Probability with Permutations and Combinations

### What Are the Odds?

### The Addition Rule for Probability

### Conditional Probability and the Multiplication Rule

### The Binomial Distribution

### Expected Value

### Chapter Test


# Statistics

## Introduction

Before the 2021 WNBA season, professional basketball player Candace Parker signed a contract with the Chicago Sky, which entitled her to a salary of $190,000. This amount was the 23rd highest in the league at the time. How did the team’s management decide on her salary? They likely considered some intangible qualities, like her leadership skills. However, much of their deliberations probably took into account her performance on the court. For example, Parker led the league in rebounds in the 2020 season (214 of them) and scored 14.7 points per game (which ranked her 18th among all WNBA players). Further, Parker brought 13 seasons of experience to the team. All of these factors played a role in deciding the terms of her contract.
Estimating the value of one variable (like salary) based on other, measurable variables (points per game, experience, rebounds, etc.) is among the most important applications of statistics, which is the mathematical field devoted to gathering, organizing, summarizing, and making decisions based on data.


# Statistics

## Gathering and Organizing Data

### Learning Objectives

After completing this section, you should be able to:
1. Distinguish among sampling techniques.
2. Organize data using an appropriate method.
3. Create frequency distributions.
When a polling organization wants to try to establish which candidate will win an upcoming election, the first steps are to write questions for the survey and to choose which people will be asked to respond to the survey. These can seem like simple steps, but they have far-reaching implications in the analysis the pollsters will later carry out. The process by which samples (or groups of units from which we collect data) are chosen can strongly affect the data that are collected. Units are anything that can be measured or surveyed (such as people, animals, objectives, or experiments) and data are observations made on units.
One of the most famous failures of good sampling occurred in the first half of the twentieth century. The Literary Digest was among the most respected magazines of the early twentieth century. Despite the name, the Digest was a weekly newsmagazine. Starting in 1916, the Digest conducted a poll to try to predict the winner of each US Presidential election. For the most part, their results were good; they correctly predicted the outcome of all five elections between 1916 and 1932. In 1936, the incumbent President Franklin Delano Roosevelt faced Kansas governor Alf Landon, and once again the Digest ran their famous poll, with results published the week before the election. Their conclusion? Landon would win in a landslide, 57% to 43%. Once the actual votes had been counted, though, Roosevelt ended up with 61% of the popular vote, 18% more than the poll predicted. What went wrong?
The short answer is that the people who were chosen to receive the survey (over ten million of them!) were not a good representation of the population of voting adults. The sample was chosen using the Digest's own base of subscribers as well as publicly available lists of people that were likely adults (and therefore eligible to vote), mostly phone books and vehicle registration records. The pollsters then mailed every single person on these lists a survey. Around a quarter of those surveys were returned; this constituted the sample that was used to make the Digest’s disastrously incorrect prediction. However, the Digest made an error in failing to consider that the election was happening during the Great Depression, and only the wealthy had disposable income to spend on telephone lines, automobiles, and magazine subscriptions. Thus, only the wealthy were sent the Digest’s survey. Since Roosevelt was extremely popular among poorer voters, many of Roosevelt’s supporters were excluded from the Digest’s sample.
Another more complicated factor was the low response rate; only around 25% of the surveys were returned. This created what’s called a non-response bias.

### Sampling and Gathering Data

The Digest's failure highlights the need for what is now considered the most important criterion for sampling: randomness. This randomness can be achieved in several ways. Here we cover some of the most common.
A simple random sample is chosen in a way that every unit in the population has an equal chance of being selected, and the chances of a unit being selected do not depend on the units already chosen. An example of this is choosing a group of people by drawing names out of a hat (assuming the names are well-mixed in the hat).
A systematic random sample is selected from an ordered list of the population (for example, names sorted alphabetically or students listed by student ID). First, we decide what proportion of the population will be in our sample. We want to express that proportion as a fraction with 1 in the numerator. Let’s call that number D. Next, we’ll choose a random number between one and D. The unit at that position will go into our sample. We’ll find the rest of our sample by choosing every Dth unit in the list, starting with our random number.
To walk through an example, let’s say we want to sample 2% of the population: . (Note: If the number in the denominator isn’t a whole number, we can just round it off. This part of the process doesn’t have to be precise.) We can then use a random number generator to find a random number between 1 and 50; let's use 31. In our example, our sample would then be the units in the list at positions 31, 81 (31 + 50), 131 (81 + 50), and so forth.
A stratified sample is one chosen so that particular groups in the population are certain to be represented. Let’s say you are studying the population of students in a large high school (where the grades run from 9th to 12th), and you want to choose a sample of 12 students. If you use a simple or systematic random sample, there’s a pretty good chance that you’ll miss one grade completely. In a stratified sample, you would first divide the population into groups (the strata), then take a random sample within each stratum (that’s the singular form of “strata”). In the high school example, we could divide the population into grades, then take a random sample of three students within each grade. That would get us to the 12 students we need while ensuring coverage of each grade.
A cluster sample is a sample where clusters of units are chosen at random, instead of choosing individual units. For example, if we need a sample of college students, we may take a list of all the course sections being offered at the college, choose three of them at random (the sections are the clusters), and then survey all the students in those sections. A sample like this one has the advantage of convenience: If the survey needs to be administered in person, many of your sample units will be located in one place at the same time.

### Organizing Data

Once data have been collected, we turn our attention to analysis. Before we analyze, though, it’s useful to reorganize the data into a format that makes the analysis easier. For example, if our data were collected using a paper survey, our raw data are all broken down by respondent (represented by an individual response sheet). To perform an analysis on all the responses to an individual question, we need to first group all the responses to each question together. The way we organize the data depends on the type of data we’ve collected.
There are two broad types of data: categorical and quantitative.  Categorical data classifies the unit into a group (or category). Examples of categorical data include a response to a yes-or-no question, or the color of a person’s eyes. Quantitative data is a numerical measure of a property of a unit. Examples of quantitative data include the time it takes for a rat to run through a maze or a person’s daily calorie intake. We’ll look at each type of data in turn when considering how best to organize.

### Categorical Data Organization

The best way to organize categorical data is using a categorical frequency distribution. A categorical frequency distribution is a table with two columns. The first contains all the categories present in the data, each listed once. The second contains the frequencies of each category, which are just a count of how often each category appears in the data.

### Quantitative Data

We have a couple of options available for organizing quantitative data. If there are just a few possible responses, we can create a frequency distribution just like the ones we made for categorical data above. For example, if we’re surveying a group of high school students and we ask for each student’s age, we’ll likely only get whole-number responses between 13 and 19. Since there are only around seven (and likely fewer) possible responses, we can treat the data as if they’re categorical and create a frequency distribution as before.
If there are many possible responses, a frequency distribution table like the ones we’ve seen so far isn’t really useful; there will likely be many responses with a frequency of one, which means the table will be no better than looking at the raw data. In these cases, we can create a binned frequency distribution. A binned frequency distribution groups the data into ranges of values called bins, then records the number of responses in each bin.
For example, if we have height data for individuals measured in centimeters, we might create bins like 150–155 cm, 155–160 cm, and so forth (making sure that every data value falls into a bin). We must be careful, though; in this scenario, it’s not clear which bin would contain a response of 155 cm. Usually, responses on the edge of a bin are placed in the higher bin, but it’s good practice to make that clear. In cases where responses are rounded off, you can avoid this issue by leaving a gap between the bins that couldn’t contain any responses. In our example, if the measurements were all rounded off to the nearest centimeter, we could make bins like 150–154 cm, 155–159 cm, etc. (since a response like 154.2 isn’t possible). We’ll use this method going forward. How do we decide what the boundaries of our bins should be? There’s no one right way to do that, but there are some guidelines that can be helpful.
1. Every data value should fall into exactly one bin. For example, if the lowest value in our data is 42, the lowest bin should not be 45–49.
2. Every bin should have the same width. Note that if we shift the upper limits of our bins down a bit to avoid ambiguity (like described above), we can’t simply subtract the lower limit from the upper limit to get the bin width; instead, we subtract the lower limit of the bin from the lower limit of the next bin. For example, if we’re looking at GPAs rounded to the nearest hundredth, we might choose bins like 2.00–2.24, 2.25–2.49, 2.50–2.74, etc. These bins all have a width of 0.25.
3. If the minimum or maximum value of the data falls right on the boundary between two bins, then it’s OK to bend the rule just a little in order to avoid having an additional bin containing just that one value. We’ll see an example of this in just a moment.
4. If we have too many or too few bins, it can be difficult to get a good sense of the distribution. Seven or eight bins is ideal, but that’s not a firm rule; anything between five and twelve is fine. We often choose the number of bins so that the widths are round numbers.

### Check Your Understanding

### Key Terms

1. sample
2. units
3. data
4. population
5. simple random sample
6. systematic random sample
7. stratified sample
8. cluster sample
9. quantitative data
10. categorical data
11. categorical frequency Distribution
12. binned frequency distribution

### Key Concepts

1. Categorical data places units into groups (categories), while quantitative data is a numerical measure of a property of a unit.
2. The sampling method for a study depends on the way that randomization is used to select units for the sample.
3. Frequency distributions help to summarize data by counting the number of units that fall into a particular category or range of quantitative values.


# Statistics

## Visualizing Data

### Learning Objectives

After completing this section, you should be able to:
1. Create charts and graphs to appropriately represent data.
2. Interpret visual representations of data.
3. Determine misleading components in data displayed visually.
Summarizing raw data is the first step we must take when we want to communicate the results of a study or experiment to a broad audience. However, even organized data can be difficult to read; for example, if a frequency table is large, it can be tough to compare the first row to the last row. As the old saying goes: a picture is worth a thousand words (or, in this case, summary statistics)! Just as our techniques for organizing data depended on the type of data we were looking at, the methods we’ll use for creating visualizations will vary. Let’s start by considering categorical data.

### Visualizing Categorical Data

If the data we’re visualizing is categorical, then we want a quick way to represent graphically the relative numbers of units that fall in each category. When we created the frequency distributions in the last section, all we did was count the number of units in each category and record that number (this was the frequency of that category). Frequencies are nice when we’re organizing and summarizing data; they’re easy to compute, and they’re always whole numbers. But they can be difficult to understand for an outsider who’s being introduced to your data.
Let’s consider a quick example. Suppose you surveyed some people and asked for their favorite color. You communicated your results using a frequency distribution. Jerry is interested in data on favorite colors, so he reads your frequency distribution. The first row shows that twelve people indicated green was their favorite color. However, Jerry has no way of knowing if that’s a lot of people without knowing how many people total took your survey. Twelve is a pretty significant number if only twenty-five people took the survey, but it’s next to nothing if you recorded a thousand responses. For that reason, we will often summarize categorical data not with frequencies, but with proportions. The proportion of data that fall into a particular category is computed by dividing the frequency for that category by the total number of units in the data.

Proportions can be expressed as fractions, decimals, or percentages.
Now that we can compute proportions, let’s turn to visualizations. There are two primary visualizations that we’ll use for categorical data: bar charts and pie charts. Both of these data representations work on the same principle: If proportions are represented as areas, then it’s easy to compare two proportions by assessing the corresponding areas. Let’s look at bar charts first.

### Bar Charts

A bar chart is a visualization of categorical data that consists of a series of rectangles arranged side-by-side (but not touching). Each rectangle corresponds to one of the categories. All of the rectangles have the same width. The height of each rectangle corresponds to either the number of units in the corresponding category or the proportion of the total units that fall into the category.
In practice, most graphs are now made with computers. You can use Google Sheets, which is available for free from any web browser.
Now that we’ve explored how bar graphs are made, let’s get some practice reading bar graphs.

### Pie Charts

 A pie chart consists of a circle divided into wedges, with each wedge corresponding to a category. The proportion of the area of the entire circle that each wedge represents corresponds to the proportion of the data in that category. Pie charts are difficult to make without technology because they require careful measurements of angles and precise circles, both of which are tasks better left to computers.
Pie charts are sometimes embellished with features like labels in the slices (which might be the categories, the frequencies in each category, or the proportions in each category) or a legend that explains which colors correspond to which categories. When making your own pie chart, you can decide which of those to include. The only rule is that there has to be some way to connect the slices to the categories (either through labels or a legend).

### Visualizing Quantitative Data

There are several good ways to visualize quantitative data. In this section, we’ll talk about two types: stem-and-leaf plots and histograms.

### Stem-and-Leaf Plots

Stem-and-leaf plots are visualization tools that fall somewhere between a list of all the raw data and a graph. A stem-and-leaf plot consists of a list of stems on the left and the corresponding leaves on the right, separated by a line. The stems are the numbers that make up the data only up to the next-to-last digit, and the leaves are the final digits. There is one leaf for every data value (which means that leaves may be repeated), and the leaves should be evenly spaced across all stems. These plots are really nothing more than a fancy way of listing out all the raw data; as a result, they shouldn’t be used to visualize large datasets.
This concept can be difficult to understand without referencing an example, so let’s first look at how to read a stem-and-leaf plot.
Stem-and-leaf plots are useful in that they give us a sense of the shape of the data. Are the data evenly spread out over the stems, or are some stems “heavier” with leaves? Are the heavy stems on the low side, the high side, or somewhere in the middle? These are questions about the distribution of the data, or how the data are spread out over the range of possible values.
Some words we use to describe distributions are uniform (data are equally distributed across the range), symmetric (data are bunched up in the middle, then taper off in the same way above and below the middle), left-skewed (data are bunched up at the high end or larger values, and taper off toward the low end or smaller values), and right-skewed (data are bunched up at the low end, and taper off toward the high end). See below figures.
Looking back at the stem-and-leaf plot in the previous example, we can see that the data are bunched up at the low end and taper off toward the high end; that set of data is right-skewed. Knowing the distribution of a set of data gives us useful information about the property that the data are measuring.
Now that we have a better idea of how to read a stem-and-leaf plot, we’re ready to create our own.
As we mentioned above, stem-and-leaf plots aren’t always going to be useful. For example, if all the data in your dataset are between 20 and 29, then you’ll just have one stem, which isn’t terribly useful. (Although there are methods like stem splitting for addressing that particular problem, we won’t go into those at this time.) On the other end of the spectrum, the data may be so spread out that every stem has only one leaf. (This problem can sometimes be addressed by rounding off the data values to the tens, hundreds, or some other place value, then using that place for the leaves.) Finally, if you have dozens or hundreds (or more) of data values, then a stem-and-leaf plot becomes too unwieldy to be useful. Fortunately, we have other tools we can use.

### Histograms

Histograms are visualizations that can be used for any set of quantitative data, no matter how big or spread out. They differ from a categorical bar chart in that the horizontal axis is labeled with numbers (not ranges of numbers), and the bars are drawn so that they touch each other. The heights of the bars reflect the frequencies in each bin. Unlike with stem-and-leaf plots, we cannot recreate the original dataset from a histogram. However, histograms are easy to make with technology and are great for identifying the distribution of our data. Let’s first create one histogram without technology to help us better understand how histograms work.
Now that we’ve seen the connection between stem-and-leaf plots and histograms, we are ready to look at how we can use Google Sheets to build histograms.
Let’s use Google Sheets to create a histogram for a large dataset.

### Bar Charts for Labeled Data

Sometimes we have quantitative data where each value is labeled according to the source of the data. For example, in the Your Turn above, you looked at in-state tuition data. Every value you used to create that histogram was associated with a school; the schools are the labels. In YOUR TURN 8.11, you found a histogram of the wins of every Major League Baseball team in 2019. Each of those win totals had a label: the team. If we’re interested in visualizing differences among the different teams, or schools, or whatever the labels are, we create a different version of the bar graph known as a bar chart for labeled data.
These graphs are made in Google Sheets in exactly the same way as regular bar graphs. The only change is that the vertical axis will be labeled with the units for your quantitative data instead of just “Frequency.”

### Misleading Graphs

Graphical representations of data can be manipulated in ways that intentionally mislead the reader. There are two primary ways this can be done: by manipulating the scales on the axes and by manipulating or misrepresenting areas of bars. Let’s look at some examples of these.

### Check Your Understanding

### Key Terms

1. proportion
2. bar chart
3. pie chart
4. stem-and-leaf plot
5. distribution (of quantitative data)
6. histogram
7. bar chart for labeled data

### Key Concepts

1. Categorical data can be visualized using pie charts or bar charts; quantitative data can be visualized using stem-and-leaf plots or histograms.
2. Areas in pie charts and bar charts represent proportions of the data falling into a particular category, while areas in histograms represent proportions of the data that fall into a given range of data values (or “bins”).  Stem-and-leaf plots are visual representations of entire datasets.
3. By manipulating the axes, changing widths of bars, or making bad choices for bins, we can create data visualizations that misrepresent the distribution of data.

### Videos

1. Make a Simple Bar Graph in Google Sheets
2. Create Pie Charts Using Google Sheets
3. Make a Histogram Using Google Sheets
4. How to Spot a Misleading Graph