Grade 7: Probability, Percent & Rational Numbers

Objective

Students will be introduced to the basic language and concepts of probability — sample space, the probability scale (impossible, unlikely, equally likely, likely, certain, or a number between 0 and 1), and the difference between experimental and theoretical probability — using real-world examples (weather, batting averages) and the historical story of Chevalier de Méré, Pascal, and Fermat.

Materials

• This resource (Chapter 1 introduction and Section 1.1 text, printed or displayed)
• Whiteboard/chart paper and markers
• Paper for students to take notes and answer questions

Warm-up (~5 min)

Write on the board: "The weather forecast says there is a 70% chance of rain today. You have outdoor plans. Should you go ahead or cancel?"

• Ask students to jot down their answer and reasoning (1–2 sentences).
• Take a few quick volunteer answers. Point out that this is an example of probability being used in everyday life — a percent is being used to describe the likelihood of something happening.
• Tell students: today's lesson is about how mathematicians began thinking about likelihood and chance, and the vocabulary we use to describe it.

Main Activity (~25 min)

Part 1: Probability in daily life (7 min)
Read aloud or have students read the passage describing real-life uses of probability:

• The weather forecast example (chance of rain)
• The batting average example: a player batting .300 has only a 30% chance of getting a hit at any at-bat; a .300 hitter is 50% more likely to get on base than a .200 hitter.
• The lottery example: many people play despite very low chances of winning.

Ask students: "In each of these examples, what does the percent or decimal actually represent?" Guide discussion toward the idea that probability is a ratio of a part to a whole (a fraction, decimal, or percent).

Part 2: Key vocabulary (5 min)
On the board, write and briefly explain these terms from the text:

• Sample space — the set of all possible outcomes of an experiment
• Probability of an event — can be described in words: impossible, unlikely, equally likely, likely, certain, or as a number between 0 and 1
• Experimental (empirical) probability vs. theoretical probability — mention that students will learn to recognize the difference and how they relate

Part 3: The story of Chevalier de Méré (13 min)
Read/summarize the historical passage aloud to the class:

• In the 15th century, a gambler named Chevalier de Méré bet he could roll at least one 6 in four rolls of a die. He reasoned (incorrectly) that since the chance of a six on one roll is 1/6, the chance in four rolls must be 4 × 1/6 = 2/3. The actual probability, the text tells us, is 1 − (5/6)⁴, or 51.8%.
• He then tried a new bet: rolling at least one double-six in 24 rolls of a pair of dice, reasoning the probability would be 24 × 1/36 = 2/3. This time he started losing money, so he asked his friend Blaise Pascal for help. Pascal and Pierre Fermat worked out the problem together — an event often marked as the beginning of the mathematical theory of probability.

Have students discuss in pairs or small groups:

1. What did de Méré assume happens to probabilities when you repeat an experiment (his mistaken idea)?
2. The text notes that if his reasoning were extended to 3 rolls, the "probability" would total 1 (certain) — but we know you could roll a die many times and never get a six. Why does this show his reasoning must be wrong?

Bring the class back together and discuss answers, reinforcing the point from the text: probabilities from repeated trials do not simply add — as Pascal realized, they multiply.

Wrap-up / Exit Ticket (~10 min)

Have students answer the following on paper, using only what was discussed in class:

1. In the batting average example, what does a .300 batting average tell us about the probability of getting a hit?
2. List the five words (plus one more description) used in the text to describe the probability of an event, ranging from 0 to 1.
3. What was the mistake in Chevalier de Méré's reasoning about rolling a six in four rolls of a die?
4. In your own words, what is the difference between experimental and theoretical probability, based on what we discussed today?

Collect the exit tickets as a quick check for understanding.

If Time Remains

Have students revisit de Méré's second bet — the double-six bet with 24 rolls of a pair of dice (where he reasoned 24 × 1/36 = 2/3). Ask them to discuss in pairs: "Based on what we learned about the first dice game, why might this same type of reasoning (multiplying and adding straight across) also be flawed here?" Have a few pairs share their reasoning with the class.