Chapter 7: Probability and Statistics

In this chapter, you will learn about data. Data is information we collect, like numbers from a survey or from rolling dice. You will learn how to gather data, make plots (pictures that show data), and compare different sets of data. You will also learn about the middle value of data (called the center) and how spread out the data is.

You will also learn about probability. Probability tells us how likely something is to happen. You will use lists, tables, and tree diagrams (branching pictures) to figure out the chances of things happening — even when two things happen together.

A Story About Statistics

Long ago, during a war called the Crimean War, a nurse named Florence Nightingale noticed something sad. Many soldiers were dying in the hospital — not from their injuries, but from something else going wrong afterward. She thought this might be because the hospitals were not clean.

She collected data from hospitals that were clean and hospitals that were not clean. She wanted to find a fix that could help right away. After studying the data, she found one important rule: doctors should wash their hands!

She told the leaders in charge, but they did not act fast enough. So she asked Queen Victoria for help. To show the Queen how important this was, she invented the bar graph — a picture made of bars that shows data clearly. It worked! The Queen ordered doctors to wash their hands. This was the start of using science and data in medicine and statistics.

A Story About Probability

Probability and statistics are connected, but they started in different ways. Probability began with games of chance, like dice games. Long ago, gamblers believed in lucky rules, even without proof.

A long time ago, a man named Chevalier de Méré asked a mathematician named Blaise Pascal about a dice game rule that wasn't working for him. Pascal talked with another mathematician, Pierre Fermat. Together, they built the beginnings of probability theory. Today, this kind of math helps us understand random events in biology, money, and more.

What You Will Learn

Part 1 starts with basic probability using things like dice and cards. You will use fractions and decimals to calculate probability. Sometimes we don't know a probability for sure, but we can guess it by trying something many times. This is called the Law of Large Numbers. You might try tossing a chocolate candy many times and counting how often it lands on its flat bottom!

Part 2 teaches you about taking random samples — small groups picked from a bigger group, called a population. Since we usually can't measure an entire population, we study a sample instead. This is called inferential statistics — using a sample to make good guesses about the whole population. Picking a sample randomly (by chance, fairly) is very important. You will collect samples, make plots, do calculations, and use what you learn to make guesses about the whole group.

Part 3 compares two or more populations. You will use real data and data you collect yourself. You will compare plots and compare the center and spread of different groups, using both math and by just looking at the pictures.

This chapter will help you understand why fairness matters when picking samples, and how we use samples to learn about bigger groups. You will also study chance events, including when two events happen together, using tables, lists, and tree diagrams.

What is a Fair Game?

A simple game has:

• Players (2 or more people)
• A tableau (the space where the game happens)
• Moves (the actions players can take)
• Outcomes (the different ways the game can end)

There is also a rule that decides who wins. A game is called fair if every outcome is equally likely to happen.

Game 1: The Two-Spinner Game

Two players each have a spinner. Each spinner is split into 5 equal parts. Each part has one of these numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. No number is on both spinners. Both players spin, and whoever gets the higher number wins.

Is this fair? Not always! If Player A has {0,1,2,3,4} and Player B has {5,6,7,8,9}, then Player B always wins. That's not fair at all.

What if Player A gets all the odd numbers, and Player B gets all the even numbers? Let's check.

Example 1: Player A has odd numbers, Player B has even numbers.

There are 5 odd numbers and 5 even numbers. If we pair every odd number with every even number, we get 25 possible outcomes (5 × 5 = 25). Since 25 cannot be split evenly into two equal groups, this game cannot be fair. But we can't tell just from this whether A or B has the better chance — sometimes one player has the edge, sometimes the other.

Example 2: Player A has {0, 2, 4, 6, 8, 10}. Player B has {1, 3, 5, 7, 9}.

This time there are 6 × 5 = 30 possible outcomes. Let's count how many times A wins:

• If A spins 0, A loses every time.
• If A spins 2, A beats only 1 of B's numbers.
• If A spins 4, A beats 3 of B's numbers.
• And so on.

Adding these up: 5 + 4 + 3 + 2 + 1 = 15 times A loses. Since there are 30 outcomes total, A must win the other 15 times. Since 15 equals 15, this game is fair!

Game 2: Player B Always Wins

In this game, there are four spinners, each with 3 sections:

• 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 spinners left. Whoever spins the higher number wins. If it's a tie, they spin again.

Let's say Player A picks Blue {4, 4, 2}, and Player B picks Yellow {6, 2, 2}. There are 9 possible outcomes (pairs of numbers). If we count them carefully:

• Player A wins 4 times.
• Player B wins 3 times.
• There are 2 ties.

Since ties just lead to another spin (and the same numbers), Player A still has more chances to win. So this game is not fair — it favors Player A in this matchup.

Example 3: Is there a spinner Player B could pick to have the better chance, no matter which spinner Player A picks first?

Yes! If Player B always picks the spinner that comes right after Player A's spinner in the list (Red, Blue, Green, Yellow, and then back to Red), Player B will have the better chance of winning. For example, if Player A picks Yellow, Player B should pick Red.