Chapter 1: Probability, Percent, and Fractions That Mean the Same Thing

Why We Start With Probability

In 7th grade, students usually start the year by reviewing everything they learned about numbers before. But this can feel boring. Many kids already know it, or they don't learn much from doing it again.

So instead, this book tries something new. We use probability as a fun new topic. Probability means figuring out how likely something is to happen. Learning about probability still lets students practice math they already know, like fractions and decimals. But it feels new and interesting, not like boring review.

Starting the year with probability helps in three ways:

1. It grabs students' attention.
2. It shows how math helps us understand the real world.
3. It helps the class talk and work together from the very start.

In this chapter, students will:

• Get better at working with fractions, percents, and decimals.
• Learn that fractions, percents, and decimals can all show the same amount, depending on what we call "the whole."
• Compare and put fractions in order (including fractions less than zero).
• Solve problems with percents, like discounts, interest, taxes, tips, and increases or decreases.

What Is Probability, and Why Does It Matter?

This is the first time students formally learn about probability. Before this, students worked with sets of information called data. In 7th grade, students start learning statistics—tools for studying data. Probability is the foundation underneath statistics.

Here's an example of how this works. Imagine doctors test a new medicine. They might say: "This medicine works for 80% of patients, but causes side effects in 4% of them." Those percentages come from probability—assigning numbers to how likely different outcomes are.

In short: probability is a part of math that supports statistics. And statistics is at the heart of the scientific method: come up with an idea, collect data, study the data, and figure out how likely the idea is true.

Section 1.1: Studying Chance and Building Probability Models

In the first section, students study chance processes. These are experiments or situations where we know all the possible outcomes, but we don't know which one will happen each time.

Students will look at probability as a comparison between "part" and "whole," shown as fractions, decimals, or percents. (Later, in Chapter 4, students will compare "part" to "part," which is different and important too.) We can always change a part-to-whole comparison into a part-to-part comparison, or the other way around. For example, if 3 out of 5 kids in a class are girls, then 2 out of 5 are boys, and the ratio of girls to boys is 3:2. Later, in Chapter 7, students will learn about odds, which compare part to part.

Probability often comes from doing experiments. All the possible results of an experiment are called the sample space. One single result is called an event. We can describe how likely an event is using words like impossible, unlikely, equally likely, likely, or certain. Or we can use a number between 0 and 1.

Students will learn about two types of probability:

• Experimental probability: based on what actually happened when you tried something.
• Theoretical probability: based on what should happen, using math and reasoning.

Students will learn how these two types are alike and different.

Probability Shows Up Everywhere

Probability is all around us in daily life.

Weather: If the weather report says there's a 70% chance of rain, should you still go on your outdoor trip, or cancel it?

Sports: A baseball player's batting average is a kind of probability. It tells us, based on past games, how likely the player is to get a hit. If a player's batting average is .300, that means they only get a hit about 30% of the time they bat!

The Lottery: Millions of people buy lottery tickets, hoping to win a lot of money. But their chances of winning are actually very, very small.

Studying probability helps students practice math with whole numbers and fractions. It also gets them ready for statistics (Chapter 7), since probability helps explain randomness—like why results can be a little different each time you repeat an experiment.

Where Did Probability Come From?

In the 1400s, mathematicians started looking for ways to measure how likely things are to happen.

Many think this study began with an unfinished dice game. A gambler named Chevalier de Méré wrote a letter to his friend, a French mathematician named Blaise Pascal. De Méré made money betting on two dice games.

Game 1: He bet he could roll at least one 6 in four rolls of a die. He correctly figured that the chance of rolling a 6 on one roll is 1/6. But he wrongly thought that in four rolls, the chance of getting a 6 would be:

4 × 1/6 = 2/3

Even though his math was wrong, he still won money over time with this bet! (Today we know the real chance of winning is about 51.8%.)

Game 2: Since people stopped betting on Game 1, de Méré changed the game. Now he bet that he could roll double 6's at least once in 24 rolls of two dice. This time, he started losing money! He correctly figured that the chance of rolling double 6's on one roll of two dice is 1/36. But he wrongly thought that in 24 rolls, his chance of getting a double 6 would be:

24/36 = 2/3

Since he kept losing money, de Méré knew something was wrong with his math. So he asked his friend Pascal for help. Pascal worked on the problem with another mathematician, Pierre Fermat. Together, they solved it. Many people say this is the beginning of probability as a math topic.

A Closer Look: Why Probabilities Don't Just Add Up

Studying this story also helps us understand two important math operations: addition and multiplication of fractions.

De Méré's mistake was in how he added fractions. He thought that if the chance of rolling a 6 once is 1/6, then the chance in two rolls would be double that (1/3), and in four rolls it would be 2/3.

He seemed to think that every time you repeat an experiment, the chances of success just add together. But this can't be right! If we followed his thinking one more time, the chance of rolling a 6 in three rolls would be:

1/3 + 1/3 + 1/3 = 1 (which means "always happens")

But that's not true. You could roll a die 3 times and never get a 6. You could even roll it 500 times and never get a 6!

So de Méré asked Pascal to explain what was wrong. Pascal realized that in cases like this, probabilities do not add together. Instead, they multiply. For example, to find the chance of rolling a 6 two times in a row, you multiply the chances together instead of adding them.