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

Expected value

Statistics and probability

CCSS HSS.MD.A.2CCSS HSS.MD.B.5

Objective and success criteria

Students will: Expected value.

  • 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: Experimental probability and expected results; Permutations and combinations

Leads to: Use expected value to judge whether a game is fair

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 (Experimental probability and expected results; Permutations and combinations). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. Sam flips a drawing pin 20 times. It lands point up 7 times. What is the relative frequency of landing point up? Give a fraction in simplest form. → 7/20Prompt: Relative frequency = number of times it happened ÷ number of trials.
  2. In how many ways can 1st, 2nd, 3rd and 4th place be awarded among 6 runners? → 360Prompt: Order matters: use permutations nPr.
  3. The probability of an event is 1/2. The experiment is done 12 times. How many times would you expect the event to happen? → 6Prompt: Expected number = probability × number of trials.

Hook · 3 min

  • Take a quick class survey (favorite fruit, travel to school) and ask what the results tell us.
  • Share the objective: Expected value. 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: A game pays 9 points with probability 0.5, 8 points with probability 0.3, 5 points with probability 0.2. What is the expected number of points?
  • Step 1: 9 × 0.5 + 8 × 0.3 + 5 × 0.2 = 7.9
  • Answer: 7.9
  • 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. A game pays 3 points with probability 0.2, 9 points with probability 0.2, 10 points with probability 0.6. What is the expected number of points? → 8.4Prompt: E(X) = Σ x × P(x).
  2. A raffle ticket costs 5 points. There is a 0.05 chance of winning 21 points. What is the expected gain per ticket (prize value minus cost)? It may be negative. → -3.95Prompt: Expected winnings = probability × prize.

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. A raffle ticket costs 5 points. There is a 0.2 chance of winning 20 points. What is the expected gain per ticket (prize value minus cost)? It may be negative. → -1Prompt: Expected winnings = probability × prize.
  2. A game pays 2 points with probability 0.2, 6 points with probability 0.2, 3 points with probability 0.6. What is the expected number of points? → 3.4Prompt: E(X) = Σ x × P(x).

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: Experimental probability and expected results.
  • 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

  • Move on to: Use expected value to judge whether a game is fair.
  • 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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Expected value

Name

Date

Score/ 6

  1. Question 1

    A game pays 3 points with probability 0.25, 4 points with probability 0.25, 10 points with probability 0.5. What is the expected number of points?

    Answer

  2. Question 2

    A raffle ticket costs 4 points. There is a 0.2 chance of winning 9 points. What is the expected gain per ticket (prize value minus cost)? It may be negative.

    Answer

  3. Question 3

    A game pays 2 points with probability 0.2, 9 points with probability 0.2, 1 points with probability 0.6. What is the expected number of points?

    Answer

  4. Question 4

    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.

    Answer

  5. Question 5

    A game pays 5 points with probability 0.2, 1 points with probability 0.2, 9 points with probability 0.6. What is the expected number of points?

    Answer

  6. Question 6

    A raffle ticket costs 2 points. There is a 0.2 chance of winning 23 points. What is the expected gain per ticket (prize value minus cost)? It may be negative.

    Answer

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Answer key: Expected value

  1. 1. 6.75

  2. 2. -2.2

  3. 3. 2.8

  4. 4. -0.4

  5. 5. 6.6

  6. 6. 2.6