Chapter 7

Probability and Statistics

In this chapter, you will learn how to collect samples of data and use them to make smart guesses (called inferences) about a larger group. You'll pay close attention to two big ideas: measures of central tendency (ways to describe the "middle" of a data set, like the average) and variability (how spread out the data is). You'll practice this by gathering samples, making graphs, showing data in different ways, and comparing different sets of sample data — building on the basic statistics skills you've learned in earlier grades.

You'll also learn to find probabilities, including for compound events (events made up of two or more simpler events happening together), using organized lists, tables, and tree diagrams to display and study these events. The activities in this chapter are designed to help you move from hands-on experience to broader ideas — general rules — about probability and numbers. You'll compare graphs of data from different groups to compare their center and spread, using both calculations and careful observation.

A Bit of History

Statistics as a science began in the mid-1800s, during the Crimean War. Florence Nightingale, a nurse serving with the British Army, noticed that many British soldiers were dying in the hospital from "complications" caused by their wounds — not from the wounds themselves. She suspected the real problem was unsanitary conditions in the field hospitals, not the injuries.

To test her idea, Nightingale collected data from hospitals that tried to stay clean and sanitary, and from those that didn't. Her goal wasn't just to find the cause of the problem — she wanted to suggest a fix that could be put into action right away. After studying the data and comparing hospital practices to death rates, she found one simple rule that could make a big difference: doctors should wash their hands.

When she brought this idea to military leaders, they didn't act with the urgency she felt was needed. So she took her case straight to Queen Victoria. To help the Queen understand how important this was, Nightingale invented the bar graph — a way of showing data as bars of different heights — so the numbers would be easy to see and understand. It worked. The Queen ordered doctors to wash their hands, and this moment marked the beginning of scientific statistics and modern medical practice.

Where Probability Came From

Although probability and statistics are closely connected today, they actually started out separately. Probability questions first came up in games of chance. For centuries, gamblers trusted rules about winning that were based on tradition, whether or not those rules actually made sense mathematically.

As you may remember from Chapter 1, in the mid-1600s a gambler named the Chevalier de Méré asked the mathematician Blaise Pascal about a dice game strategy that wasn't working out for him. Pascal began writing letters back and forth with another brilliant mathematician, Pierre Fermat, and together they built the foundations of probability theory — all because of de Méré's bad luck at dice! Today, that theory is essential for studying many kinds of processes, especially in biology and economics, wherever random chance can affect a sequence of events.

What You'll Learn in This Chapter

Section 1 starts with basic probability ideas and notation, using objects like dice and cards. You'll build strategies for making sense of different situations, then use those strategies to form general rules. To do the necessary calculations, you'll work with fractions and decimals, which will strengthen your skills with rational numbers (numbers that can be written as fractions).

Some probabilities can't be calculated exactly, but they can be estimated by repeating an experiment many times and observing the results. This idea is called the Law of Large Numbers — the more times you repeat a trial, the closer your estimate gets to the true probability. You'll explore this by tossing a Hershey's Kiss candy many times and calculating how often it lands on its flat base.

Section 2 explores how to gather random samples in order to learn about a whole population — this is the basic idea behind inferential statistics. Usually, we can't know everything about a huge population, because it's too large or too hard to measure directly. "Inferential statistics" means collecting samples from the population and then using them to make educated judgments about the whole population.

The key to getting a sample that truly represents the population is to select it randomly. This is trickier than it sounds, and a big part of this chapter is thinking carefully about what "random sample" really means. You'll gather samples from real and imaginary populations, graph the data, do calculations on your results, and use what you find to make decisions about the larger population.

Section 3 uses inferential statistics to compare two or more populations. You'll work with existing data as well as data you collect yourself. You'll compare graphs from different populations, then compare their center and spread using both calculations and visual comparison.

This chapter highlights why fairness matters when choosing a random sample, and why samples are useful for drawing conclusions about populations. You'll practice and build on statistical tools from 6th grade, using measures of center and spread to compare populations. You'll also investigate random (chance) processes by creating, testing, and evaluating probability models. Compound events will be explored through simulations and shown in multiple ways — tables, lists, and tree diagrams.

In 8th grade, statistics will focus on scatter plots of data with two related variables (called bivariate data). This topic also appears in Secondary Math I. Later, Secondary Math I, II, and III will return to center and spread, random probability calculations, sampling, and making inferences.

Getting Started: Is It a Fair Game?

The student workbook begins with an anchor activity 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 whether a game seems unfair. It also shows that solving this kind of problem isn't always as simple as it looks at first. This same lesson comes up again later in Section 1, in a famous puzzle called the Monty Hall problem. For now, this activity introduces all the big ideas of the chapter. Let's look at this type of problem using simple games.

What makes a game fair?

A simple game has:

• players (two or more people),
• a tableau — the field or board the game is played on,
• moves — the actions players can take,
• outcomes — the possible end results,
• and a rule that decides the winner based on the outcome.

Using language from Chapter 1: for each player, the statement "this player is the winner" is called an event. All the possible outcomes can be sorted into groups based on which player wins. (Note: in many casino games, each player bets on a specific event — like "red" or "even" on a roulette wheel. Since these events can overlap, those games are not "simple" — they're called compound games.)

A game is called fair if all the outcomes are equally likely to happen. You saw an example of this with dice games in Chapter 1. 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, and each section shows one of the digits 0–9. No digit appears on both spinners. To play, both players spin at once, and whoever spins the higher number wins.

Is this a fair game? Not necessarily! If Player A's spinner has {0, 1, 2, 3, 4} and Player B's has {5, 6, 7, 8, 9}, then Player B always wins — that's clearly unfair. So is there some way to set up the spinners so the game is fair? For instance, what if Player A gets all the odd digits and Player B gets all the even digits?

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

Each outcome is a pair of numbers (a, b), where a is what Player A's spinner lands on and b is what Player B's spinner lands on. If a > b, that outcome belongs to "A wins." If a < b, it belongs to "B wins."

Example 1

Suppose Player A's spinner has all the odd digits, and Player B's spinner has all the even digits. Is this a fair game? Is there any way to arrange the numbers so the game is fair?

Solution: Every possible outcome is a pair (a, b), where a is odd and b is even. Since there are 5 odd digits and 5 even digits, there are 5 × 5 = 25 possible pairs. Since 25 is an odd number, it can't be split evenly into two equal groups — so this can't be a fair game.

But could some other arrangement of odd/even digits create a fair game? No, for the same reason: with 25 total outcomes, you can never split them into two perfectly equal groups. Sometimes Player A might have a slight edge, and sometimes Player B might — but it will never be perfectly even.

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 — an even number, so a fair game is possible. Let's count how many outcomes result in a win for Player A (meaning a > b):

• If A spins 0, A loses every time (0 pairs won).
• If A spins 2, A beats only B's 1 — but wait, let's count carefully: A beats B's numbers that are less than A's spin.
• If A spins 4, A beats B's {1, 3} — 2 wins.
• If A spins 6, A beats B's {1, 3, 5} — 3 wins.
• If A spins 8, A beats B's {1, 3, 5, 7} — 4 wins.
• If A spins 10, A beats all of B's numbers — 5 wins.

Adding these up: 0 + 1 + 2 + 3 + 4 + 5 = 15 wins for Player A. Since there are 30 total outcomes, that leaves 15 outcomes where Player B wins. Since both players win exactly 15 times, this is a fair game.

Game 2: Player B Always Wins

In this game, there are 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 players spin, and whoever gets the higher number wins. If there's a tie, they spin again.

Let's check one possible choice: suppose 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 3 × 3 = 9 possible outcomes total.

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

Since ties just lead to another spin under the same odds, they don't change the overall pattern — Player A already wins more often than Player B. So this is not a fair game.

Example 3

Once Player A has chosen a spinner, is there a particular spinner Player B could choose to have better odds of winning? (Hint: we wouldn't ask this question if the answer were "no.")

Solution: Yes! Notice the order the spinners are listed in: Red, Blue, Green, Yellow. Player B has better odds if B always picks the spinner that comes right before the one Player A picked, going around this list in a loop. (This means, for example, if Player A picks Yellow, Player B should pick Green — or looking at it another way, whatever spinner comes just before A's choice in the list gives B the edge.)