Chapter 7: Probability and Statistics

In this chapter, you will learn how to collect data samples and use them to make smart guesses (called inferences) about bigger groups. You'll learn about two things: the "middle" of a data set (called central tendency) and how spread out the data is (called variability). You'll do this by collecting samples, making graphs, showing data in different ways, and comparing different sets of sample data. This builds on the basic data skills you've learned in past years.

You'll also learn to find probabilities—including probabilities for "compound events" (events made up of two or more things happening together). You'll use organized lists, tables, and tree diagrams to show and study these events. These activities will help you move from hands-on experiments to general ideas about probability and numbers. You'll compare graphs of data from different groups to see how their centers and spreads are alike or different, using both math and careful looking.

The History of Statistics

Statistics (the study of data) began in the mid-1800s during the Crimean War. A nurse named Florence Nightingale, who worked with the British Army, noticed that many soldiers were dying in the hospital from "complications" caused by their injuries—not from the injuries themselves. She thought this might be happening because the field hospitals weren't clean, not because of how badly the soldiers were hurt.

She started collecting data from hospitals that tried to stay clean and hospitals that didn't. Her goal wasn't just to find the cause of the problem—she wanted to find a fix that could be used right away. After studying the data, she found a connection between hospital cleanliness and how many patients died. She discovered one simple rule that could make a big difference: doctors should wash their hands.

When she told the army leaders this, they didn't act quickly enough for her liking. So she went straight to Queen Victoria. To help the Queen understand how important this was, Nightingale invented the bar graph! It worked—the Queen ordered doctors to wash their hands. This moment marked the beginning of scientific statistics and modern medicine.

The History of Probability

Probability and statistics are closely connected, but they started in different ways. Probability (the study of chance) grew out of games of chance. For hundreds of years, gamblers followed rules about winning and losing—some based on real math, some just based on old beliefs.

As you may remember from Chapter 1, in the mid-1600s a man named the Chevalier de Méré asked the mathematician Blaise Pascal about a dice game rule that wasn't working out for him. Pascal then wrote letters back and forth with another mathematician, Pierre Fermat. Together, these two brilliant thinkers built a theory of probability that explained why de Méré's rule had failed. Today, that theory is a key part of studying many things, especially in biology and economics, wherever random chance plays a role.

What You'll Learn in Each Section

Section 1 starts with basic probability ideas and symbols, using objects like dice and cards. You will build strategies to understand different situations and turn them into general rules. To do the probability math, you'll practice using fractions and decimals, which will strengthen your skills with rational numbers (numbers that can be written as fractions).

Some probabilities can't be known for sure, but can be estimated by repeating a test many times. This is called the Law of Large Numbers—the more times you repeat something, the closer your results get to the true probability. You'll test this by tossing a Hershey's Kiss chocolate many times and calculating how often it lands on its flat base.

Section 2 explores how to gather random samples to learn about a whole population (a large group). This is the basis of inferential statistics—using a sample to make judgments about a bigger group. Usually, we can't measure an entire population because it's too big or too hard to study completely. That's why we use samples instead. The key is to choose samples randomly, though that's not always as easy as it sounds—this chapter will help you think carefully about what "random sample" really means. You'll gather samples from real and pretend populations, graph the data, do calculations, and use your results to make decisions about the larger population.

Section 3 uses inferential statistics to compare two or more populations. You'll use existing data and collect your own. You'll compare graphs from different populations and look at their centers and spreads, using both calculations and visual comparisons.

This chapter shows why fairness matters when picking random samples, and why samples are useful for learning about populations. You'll also practice and build on statistics skills from Grade 6, working more with measures of center and spread. You'll explore chance events by building, using, and testing probability models. You'll study compound events through simulations and by using tables, lists, and tree diagrams.

In 8th grade, you'll study scatter plots using data with two variables (called bivariate data). This topic continues into Secondary Math I. Later math courses (Secondary Math I, II, and III) return to center and spread, probability calculations, sampling, and making inferences.

Getting Started: "Teacher Always Wins!"

Your workbook begins with a game called "Teacher Always Wins!" This activity gets you thinking about what kind of data you need to solve a problem—in this case, figuring out if the game seems unfair. It also shows that solving this kind of problem isn't always as simple as it looks at first. You'll see this idea again later in the "Monty Hall problem." This starting game introduces all the big ideas of this chapter. Let's first look at this kind of problem using simple games.

What is a Fair Game?

A "simple game" has:

• Players (2 or more people)
• A tableau — the playing field where the game happens
• Moves — the actions players can take
• Outcomes — the possible ending results

Finally, there's a rule that decides the winner for each outcome. Using the language from Chapter 1, for each player A, the statement "A is the winner" is called an event. All the possible outcomes can be divided up based on which player wins.

In many casino games, each player picks their own winning event by placing a bet (like betting on "red" or "even" at a roulette table). Since these bets can overlap, this makes the game compound rather than simple.

A game is called fair if all the outcomes are equally likely to happen. Rolling dice, which we talked about in Chapter 1, is one example. Now let's look at another example: spinner games.

Game 1: The Two-Spinner Game

This is a game for two players. Each player has a spinner divided into five equal sections. Each section has one number from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, and no number appears on both spinners. Each turn, both players spin, and whoever gets the higher number wins.

Is this a fair game? Not always! For example, if Player A has {0, 1, 2, 3, 4} and Player B has {5, 6, 7, 8, 9}, Player B will always win. So is there an arrangement of numbers that makes the game fair? What if Player A gets all the odd numbers, and Player B gets all the even numbers?

To figure this out, we list every possible outcome and split them into two groups: outcomes where A wins, and outcomes where B wins. If both groups have the same number of outcomes, the game is fair.

An outcome is written as a pair of numbers (a, b), where a is the number Player A's spinner lands on, and b is the number Player B's spinner lands on. If a is bigger than b, that outcome goes in A's winning group. If b is bigger, it goes in B's winning group.

Example 1:
Suppose Player A's spinner has all the odd numbers, and Player B's spinner has all the even numbers. Is this fair? Is there any arrangement that would be fair?

Solution: Every possible outcome is a pair (a, b), where a is odd and b is even. Since there are 5 odd numbers and 5 even numbers, there are 5 × 5 = 25 total possible pairs. Since 25 is an odd number, we can't split it evenly into two equal groups. So this game can't be fair—there's no way to make the two winning groups the same size. This also means it's impossible to arrange the numbers into odds vs. evens and get a fair game, though sometimes A might have the advantage, and sometimes B might.

Example 2:
Suppose Player A's spinner has 6 sections marked {0, 2, 4, 6, 8, 10}, and Player B's spinner has 5 sections marked {1, 3, 5, 7, 9}. Is this a fair game?

Solution: This time, there are 6 × 5 = 30 possible outcomes, so a fair game is possible. Let's count how many times A wins. If A spins a 0, A loses every time. If A spins a 2, A loses to 4 of B's numbers (3, 5, 7, 9) but wins against... let's count carefully: A loses when A's number is smaller than B's. Adding up all the times A loses: 5 + 4 + 3 + 2 + 1 = 15 outcomes. Since there are 30 total outcomes, that means A also wins in 15 outcomes. Since both players win the same number of times, this is a fair game!

Game 2: Player B Always Wins

This game uses four spinners—Red, Blue, Green, and Yellow—each divided into three sections:

• Red: {3, 3, 3}
• Blue: {4, 4, 2}
• Green: {5, 5, 1}
• Yellow: {6, 2, 2}

First, Player A picks a spinner. Then Player B picks one of the remaining spinners. Both spin, and whoever gets the higher number wins. If there's a tie, they spin again.

Let's look at one example: Player A picks Blue {4, 4, 2}, and Player B picks Yellow {6, 2, 2}. Since some numbers repeat, let's label them to keep track: Blue has {4₁, 4₂, 2}, and Yellow has {6, 2₁, 2₂}. There are 9 possible outcomes total. Let's count them:

• A wins: (4₁, 2₁), (4₁, 2₂), (4₂, 2₁), (4₂, 2₂) — that's 4 outcomes
• B wins: (4₁, 6), (4₂, 6), (2, 6) — that's 3 outcomes
• Tie: (2, 2₁), (2, 2₂) — that's 2 outcomes

Even when there's a tie and they spin again, the same pattern repeats, so A still wins more often than B. This means the game is not fair—it favors Player A when this particular pair of spinners is chosen.

Example 3:
Once Player A picks a spinner, is there a choice Player B can make to have the advantage instead? (Hint: we wouldn't ask this unless the answer were "yes.")

Solution: Look closely at the game's name for a clue! It turns out Player B has the advantage if B always picks the spinner listed right after A's spinner in the list above. For example, if A picks Yellow, B should pick Red.