Chapter 7: Probability and Statistics

In this chapter, students build an understanding of how to sample data and draw conclusions from it, paying attention to both measures of central tendency (ways of describing a "typical" value, like the mean or median) and variability (how spread out the data is). Students will gather samples, create plots, represent data in various ways, and compare data sets—building on the basic statistics skills they developed in earlier grades. They will also calculate probabilities, including for compound events (situations made up of multiple simpler events), using organized lists, tables, and tree diagrams. These activities are designed to help students move from hands-on experience toward general conjectures (educated guesses) about probability and numbers. Students will also compare graphical representations of data from different populations, judging center and spread both through calculation and by eye.

A Historical Note: Where Statistics Began

Statistics as a field began in the mid-nineteenth century during the Crimean War, when Florence Nightingale—a nurse serving with the British Army—noticed that an unusually high number of British soldiers were dying in the hospital from "complications from their injuries" rather than from the injuries themselves. Nightingale suspected the real cause was the lack of sterile (germ-free) conditions in the field hospitals, not the severity of the wounds. She began collecting data from hospitals that tried to maintain sanitary conditions and from those that didn't. Her goal wasn't just to identify the cause—she wanted to propose a fix that could be put into action immediately.

After studying the data and connecting hospital practices to patient death rates, she concluded that one simple rule, if followed consistently, would make a major difference: physicians should wash their hands. When she brought this recommendation to military leadership, they didn't treat it with the urgency she felt it deserved. So she appealed directly to Queen Victoria—and to make her point clearly, she invented the bar graph so the Queen could see just how significant her findings were. It worked: the Queen ordered physicians to wash their hands, marking the beginning of scientific statistics and modern medical practice.

Where Probability Began

Probability and statistics are closely connected today, but they have very different historical roots. Probability questions arose naturally from games of chance, and for centuries gamblers relied on rules of thumb—some sound, some not—that had simply become folklore. As mentioned in Chapter 1, in the mid-seventeenth century a gambler known as the Chevalier de Méré asked the mathematician Blaise Pascal about a dice strategy that wasn't working out in his favor. Pascal began corresponding with Pierre Fermat, and together—two of the era's leading mathematicians—they developed a theory of probability that explained exactly why de Méré kept losing. Today, that theory is fundamental to studying many processes, especially in biology and economics, where random influences affect the sequence of events.

What's in This Chapter

Section 1 begins with an exploration of basic probability and notation using familiar objects like dice and cards. Students will build modeling strategies to make sense of different situations, then generalize from them. Because probability calculations require working with fractions and decimals, these exercises will also strengthen students' skills with rational number operations. Some probabilities can't be calculated directly but can be estimated by repeating a trial many times and observing the results. This is known as the Law of Large Numbers—the idea that the more times you repeat an experiment, the closer your results get to the true probability. Students will explore this by tossing a Hershey's Kiss many times and calculating how often it lands on its base.

Section 2 covers the basics of gathering random samples to learn about the characteristics of a population—in other words, the fundamentals of inferential statistics. Since most populations are too large or too difficult to measure directly, their true values usually can't be known outright. "Inferential statistics" means collecting samples from a population and analyzing them to make judgments about the population as a whole. The key to a useful sample is selecting it randomly—which is trickier than it sounds. A major goal of this chapter is to think carefully about what "random sample" really means. Students will collect samples from real and hypothetical populations, plot the data, run calculations on the results, and use that information to draw conclusions about the population.

Section 3 uses inferential statistics to compare two or more populations. Students will work with existing data sets as well as data they collect themselves, comparing plots from different populations and judging center and spread through both calculation and visual comparison.

This unit emphasizes the importance of fairness in random sampling and of using samples to make inferences about populations. Statistical tools introduced in sixth grade will be practiced and expanded as students continue working with measures of center and spread to compare populations. Students will also investigate random processes by developing, using, and evaluating probability models, and they'll explore compound events through simulations and multiple representations such as tables, lists, and tree diagrams.

Looking ahead, eighth-grade statistics will focus on scatter plots of bivariate data (data involving two variables). Bivariate data also appears in Secondary Math I. Meanwhile, Secondary Math I, II, and III revisit center and spread, random probability calculations, and sampling and inference in greater depth.

Getting Started: "Teacher Always Wins!"

The student workbook opens with an anchor activity called "Teacher Always Wins!" This game is designed to get students thinking about what kind of data is needed to solve a problem—in this case, figuring out whether the game is actually unfair. It also shows that solving such problems isn't always as simple as it first appears. This same lesson reappears near the end of Section 1 with the famous Monty Hall problem. In the meantime, this opening activity introduces every major idea in the chapter. To see these ideas in action, let's look at probability in the context of some simple games.

What Is a Fair Game?

A "simple game" has the following parts:

• Players (two or more)
• A tableau: the field or setup on which the game is played
• Moves: the actions players can take on the tableau
• Outcomes: the possible end results of the game

Finally, a rule determines the winner—it assigns one player as the winner for each possible outcome. Using the language from Chapter 1, for each player A, the statement "A is the winner" is called an event. The full set of outcomes can be split up ("partitioned") into these winning events for each player.

Note that in many casino games, each player essentially chooses their own "winning event" by placing a bet (like betting on "red" or "even" at a roulette table). In these cases, since different players' winning events can overlap, the game isn't "simple" but compound.

A game is called fair if all outcomes are equally likely. Rolling dice, discussed in Chapter 1, is one example. Below, we explore another: spinner games.

Game 1: The Two-Spinner Game

This is a game for two players. Each player has a spinner divided into five equal sectors, and each sector is labeled with one of the digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}—with no digit appearing on both spinners. Each move is a spin by both players, and whoever spins the higher number wins.

Is this a fair game? Not necessarily. For example, if Player A has {0, 1, 2, 3, 4} and Player B has {5, 6, 7, 8, 9}, Player B wins every single time. So is there an arrangement that is fair? For instance, what if Player A gets all the odd digits and Player B gets all the even digits?

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

An outcome of this game is 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 > b, that outcome belongs to A's winning set; if a < b, it belongs to B's.

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 arrangement that would make it fair?

Solution. The full set of outcomes includes every 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. Because 25 is an odd number, it cannot be split evenly into two groups of the same size—so this cannot be a fair game.

The same reasoning shows that no arrangement of odd digits versus even digits can produce a fair game—sometimes A will have the edge, and sometimes B will, but it will never be exactly even.

Example 2. Suppose Player A's spinner has 6 sectors, labeled {0, 2, 4, 6, 8, 10}, and Player B's spinner has 5 sectors, labeled {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 at least possible. Player A wins whenever a > b. If a = 0, A loses to all 5 of B's possible spins. If a = 2, A loses to 4 of B's spins (3, 5, 7, 9). If a = 4, A loses to 3 of B's spins, and so on. Adding these up: 5 + 4 + 3 + 2 + 1 = 15 outcomes where A loses. Since there are 30 total outcomes, A must win the other 15—so wins are split evenly, and 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 sectors labeled as follows:

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

Player A picks a spinner first; then Player B picks one of the remaining spinners. Both players spin, and whoever gets the higher number wins (ties are re-spun).

Let's analyze one specific choice: suppose A picks Blue {4, 4, 2} and B picks Yellow {6, 2, 2}. Since Blue and Yellow each have repeated numbers, let's label them individually to keep track: Blue has {4₁, 4₂, 2}, and Yellow has {6, 2₁, 2₂}. There are 3 × 3 = 9 total outcomes:

• 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 a tied spin just gets re-spun under the same conditions, the outcome pattern repeats—meaning A will keep winning more often than B in the long run. So this particular matchup is not a fair game.

Example 3. Once Player A has chosen a spinner, is there a specific choice B could make that favors B winning? (Hint: we wouldn't ask if the answer weren't "yes.")

Solution. Look closely at the order the spinners are listed: Red, Blue, Green, Yellow. The odds favor Player B if B always picks the spinner listed directly before the one A picked, cycling back to the end of the list if needed. In other words, if A picks Yellow, B should pick Green; if A picks Red, B should pick Yellow (looping back around the list).