Back to the lesson plan builder
Lesson length30 min45 min60 min
Questions🔀 New questions

Print the plan for yourself; the student sheet prints on its own page after it.

Math lesson plan · 12th grade (Year 13 in England) · 30 minutes

Use expected value to judge whether a game is fair

Statistics and probability

CCSS HSS.MD.B.5CCSS HSS.MD.B.7England: A-level Statistics

Objective and success criteria

Students will: Use expected value to judge whether a game is fair.

  • I can answer questions on today's skill on my own, without help.
  • I can explain each step and say why it works.
  • I can check my answer and spot a mistake.

Before and after

Needs first: Expected value

Leads to: the next skill in this area.

Materials: Squared paper for charts, Class survey data, Sticky notes for a live chart, Mini whiteboards and pens, Printed student sheet (below).

Warm-up · 4 min

  • Quick retrieval from earlier learning (Expected value). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. A game pays 2 points with probability 0.1, 4 points with probability 0.4, 6 points with probability 0.5. What is the expected number of points? → 4.8Prompt: E(X) = Σ x × P(x).
  2. A raffle ticket costs 2 points. There is a 0.1 chance of winning 16 points. What is the expected gain per ticket (prize value minus cost)? It may be negative. → -0.4Prompt: Expected winnings = probability × prize.
  3. A game pays 0 points with probability 0.1, 2 points with probability 0.4, 3 points with probability 0.5. What is the expected number of points? → 2.3Prompt: E(X) = Σ x × P(x).

Hook · 3 min

  • Take a quick class survey (favorite fruit, travel to school) and ask what the results tell us.
  • Share the objective: Use expected value to judge whether a game is fair. Read the success criteria together.

I do: model it · 7 min

  • Work through the example out loud, saying why you do each step, not just what you do.
  • Example: It costs $1 to play. You roll a fair die and win $6 if you roll a 5 or a 6. (Your stake is not returned.) What is the expected gain per game? Give the amount as a number; negative means a loss.
  • Step 1: 2/6 × 6 − 1 = 1
  • Answer: 1
  • Pause after each step and ask a student to say what comes next.

We do: guided practice · 6 min

  • Do these together. Students try each step on whiteboards before you reveal it.
  • Use the hints as prompts rather than giving the answer.
  1. It costs $3 to play. You roll a fair die and win $9 if you roll a 5 or a 6. (Your stake is not returned.) What is the expected gain per game? Give the amount as a number; negative means a loss. → 0Prompt: Expected winnings = probability × prize.
  2. A charity game costs 1 to play. The chance of winning is 0.2 and the prize is 5 (the stake is kept). Is the game fair? (Fair / Favours the player / Favours the organiser) → FairPrompt: A game is fair when the expected gain is 0.

You do: independent practice · 7 min

  • Students work on the student sheet. Circulate and check the first two answers of every student early.
  • Students who finish: write a question of their own for a partner, with the answer.

The 6 questions are on the student sheet below, with answers in the key.

Exit ticket · 3 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. It costs $1 to play. You roll a fair die and win $3 if you roll a 6. (Your stake is not returned.) What is the expected gain per game? Give the amount as a number; negative means a loss. → -0.5Prompt: Expected winnings = probability × prize.
  2. A charity game costs 1 to play. The chance of winning is 0.2 and the prize is 6 (the stake is kept). Is the game fair? (Fair / Favours the player / Favours the organiser) → Favours the playerPrompt: A game is fair when the expected gain is 0.

Watch out for

  • Forgetting to put data in order before finding the median.
  • Reading a bar's height wrongly when the scale goes up in 2s or 5s.

Support and stretch

Support

  • Review first: Expected value.
  • Use concrete materials: squared paper for charts.
  • Give the first step of each question, or a partly worked example to finish.
  • Pair with a confident partner for the guided practice.

Stretch

  • Explain a common mistake in writing and how to fix it.
  • Write a word problem that uses today's skill, then solve a partner's.

Homework and more practice

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-12.

Free lesson plan from eurekami.com/lesson-plan-builder

eurekami

Use expected value to judge whether a game is fair

Name

Date

Score/ 6

  1. Question 1

    A charity game costs 2 to play. The chance of winning is 0.2 and the prize is 11 (the stake is kept). Is the game fair?

    • A. Fair
    • B. Favours the player
    • C. Favours the organiser

  2. Question 2

    It costs $1 to play. You roll a fair die and win $9 if you roll a 5 or a 6. (Your stake is not returned.) What is the expected gain per game? Give the amount as a number; negative means a loss.

    Answer

  3. Question 3

    A charity game costs 5 to play. The chance of winning is 0.5 and the prize is 10 (the stake is kept). Is the game fair?

    • A. Fair
    • B. Favours the player
    • C. Favours the organiser

  4. Question 4

    It costs $3 to play. You roll a fair die and win $9 if you roll a 5 or a 6. (Your stake is not returned.) What is the expected gain per game? Give the amount as a number; negative means a loss.

    Answer

  5. Question 5

    It costs $3 to play. You roll a fair die and win $6 if you roll a 6. (Your stake is not returned.) What is the expected gain per game? Give the amount as a number; negative means a loss.

    Answer

  6. Question 6

    A charity game costs 3 to play. The chance of winning is 0.25 and the prize is 10 (the stake is kept). Is the game fair?

    • A. Fair
    • B. Favours the player
    • C. Favours the organiser

Free from eurekami.com

for the adult only

Answer key: Use expected value to judge whether a game is fair

  1. 1. B. Favours the player

  2. 2. 2

  3. 3. A. Fair

  4. 4. 0

  5. 5. -2

  6. 6. C. Favours the organiser