Math lesson plan · 12th grade (Year 13 in England) · 60 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 · 6 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.

A game pays 6 points with probability 0.25, 4 points with probability 0.25, 7 points with probability 0.5. What is the expected number of points? → 6Prompt: E(X) = Σ x × P(x).
A raffle ticket costs 1 points. There is a 0.1 chance of winning 26 points. What is the expected gain per ticket (prize value minus cost)? It may be negative. → 1.6Prompt: Expected winnings = probability × prize.
A game pays 7 points with probability 0.1, 3 points with probability 0.4, 6 points with probability 0.5. What is the expected number of points? → 4.9Prompt: E(X) = Σ x × P(x).

Hook · 6 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 · 12 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 · 12 min

Do these together. Students try each step on whiteboards before you reveal it.
Use the hints as prompts rather than giving the answer.

It costs $2 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. → 0Prompt: Expected winnings = probability × prize.
A charity game costs 5 to play. The chance of winning is 0.25 and the prize is 20 (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 · 17 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 8 questions are on the student sheet below, with answers in the key.

Exit ticket · 7 min

Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.

A charity game costs 5 to play. The chance of winning is 0.1 and the prize is 51 (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.
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.

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

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Use expected value to judge whether a game is fair

Name

Date

Score/ 8

1
Question 1
A charity game costs 1 to play. The chance of winning is 0.5 and the prize is 3 (the stake is kept). Is the game fair?
- A. Fair
- B. Favours the player
- C. Favours the organiser
Question 2
It costs $1 to play. You roll a fair die and win $9 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
3
Question 3
A charity game costs 4 to play. The chance of winning is 0.1 and the prize is 41 (the stake is kept). Is the game fair?
- A. Fair
- B. Favours the player
- C. Favours the organiser
Question 4
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.
Answer
5
Question 5
A charity game costs 4 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
Question 6
It costs $2 to play. You roll a fair die and win $3 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
7
Question 7
A charity game costs 3 to play. The chance of winning is 0.2 and the prize is 13 (the stake is kept). Is the game fair?
- A. Fair
- B. Favours the player
- C. Favours the organiser
Question 8
It costs $3 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.
Answer

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