Grade 7: Probability and Statistics

Objective

Students will understand what makes a game "fair" — that a game is fair when all outcomes (or all players' winning outcomes) are equally likely — and will practice this idea by analyzing spinner games using organized outcome lists, following the worked examples from Chapter 7 of Probability and Statistics.

Materials

• "Grade 7: Probability and Statistics" resource (Chapter 7 excerpt, Section 1 introduction and spinner game examples)
• Whiteboard or chart paper
• Paper and pencil for each student
• (Optional) Simple hand-drawn spinners with 5 or 6 sectors, or a paperclip and pencil to act as a spinner

Warm-up (~5 min)

1. Write on the board: "What is a fair game?"
2. Read aloud (or paraphrase) the resource's definition: a game has players, a tableau, moves, and outcomes, and a rule that decides the winner. A game is fair if all the outcomes are equally likely.
3. Ask students to turn to a neighbor and give one guess: "Do you think a game where one player wins with odd numbers and the other wins with even numbers is automatically fair? Why or why not?"
4. Take 2–3 quick verbal answers (no need to resolve yet — this sets up the main activity).

Main Activity (~25 min)

Work through the resource's spinner-game examples as a class, having students follow along on paper.

1. Introduce the Two-Spinner Game (~5 min): Explain the setup — two players each spin a spinner with 5 sectors, numbers 0–9 split between the two spinners with no repeats. Higher number wins. Ask: "If Player A has {0,1,2,3,4} and Player B has {5,6,7,8,9}, who always wins?" Confirm: Player B always wins — this is not fair.

1. Work through Example 1 together (~8 min):
2. Player A has all odd digits {1,3,5,7,9}; Player B has all even digits {0,2,4,6,8}.
3. Have students help count: there are 5 odd digits and 5 even digits, so there are 5 × 5 = 25 total outcome pairs (a,b).
4. Ask: "Can 25 outcomes be split evenly into two equal groups?" (No — 25 is odd.)
5. Conclusion from the resource: since 25 cannot be split into two equal sets, this cannot be a fair game.

1. Work through Example 2 together (~8 min):
2. Player A has {0,2,4,6,8,10} (6 sectors); Player B has {1,3,5,7,9} (5 sectors).
3. Total outcomes = 6 × 5 = 30.
4. Walk through the counting on the board exactly as in the resource: if a=0, A loses all 5 times; if a=2, A loses 4 times; if a=4, A loses 3 times; and so on, giving A loses 5+4+3+2+1 = 15 times.
5. Since there are 30 total outcomes and A loses 15 times, A also wins 15 times — so A wins half the time.
6. Conclusion: this is a fair game.

1. Quick discussion (~4 min): Ask students why counting all the outcomes carefully (not just guessing) was necessary to answer the fairness question in both examples.

Wrap-up / Exit Ticket (~10 min)

Have students answer the following on paper individually (based directly on the resource's ideas):

1. In your own words, what does it mean for a game to be "fair"?
2. In Example 1 (odd digits vs. even digits), why couldn't the game be fair? (Hint: think about the total number of outcomes.)
3. In Example 2, Player A had 6 numbers and Player B had 5 numbers, yet the game turned out to be fair. Explain how that is possible.

Collect the papers as an exit ticket.

If Time Remains

Introduce the "Second Game: Player B Always Wins" setup from the resource:

• Four spinners: 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 from what's left.
• Pose the question from the resource (Example 3): "Once Player A has picked a spinner, is there a spinner Player B could pick to have the advantage?"
• Give students the hint from the resource: the answer involves B always choosing the spinner listed directly after A's choice in the order Red, Blue, Green, Yellow (wrapping back to Red after Yellow). Let students discuss briefly why this might give B an edge, without requiring a full proof.