Lesson Plan

Odds and Expected Value

Objectives

This lesson will connect probability with the concept of odds. It is going to connect students’ current knowledge of probability and how to relate it to odds and “expected value.” Students will:

  • convert from odds to probability and vice versa.

  • calculate expected value of an experiment and the actual value.

Essential Questions

  • What differentiates an independent event from a dependent event and how are the probabilities of each calculated?

Vocabulary

  • Odds: The ratio of favorable outcomes to unfavorable outcomes in an event.

  • Expected Value: The product of the probability of a favorable outcome of an event and the quantity that represents a value associated with that favorable outcome.

  • Independent Event: Two events in which the outcome of one event does not affect the outcome of the other.

  • Dependent Event: Two events in which the outcome of one event affects the outcome of the other.

  • Probability:The mathematical probability of an event is represented by a real number, p, such that , where an impossible event is 0 and a certain event is 1. [IS.1 - Preparation]

Duration

90–120 minutes [IS.2 - All Students]

Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.

Materials

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Related Materials & Resources

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Formative Assessment

  • View
    • Teacher observations during group activities and class discussion give the teacher direct and useful information on student engagement. [IS.12 - All Students]

    • Lesson 3 Exit Ticket activity (M-A2-1-3_Lesson 3 Exit Ticket.doc and M-A2-1-3_Lesson 3 Exit Ticket KEY.doc) provides a record of individual student performance in determining and comparing probabilities. [IS.13 - All Students]

Suggested Instructional Supports

  • View
    Active Engagement, Explicit Instruction

    [IS.11 - All Students]

    W:

    Students learn to convert between probability and odds and calculate expected value of an experimental or theoretical outcome.

    H:

    The game show simulation, Wheel of Fortune, uses a familiar and relevant scenario to engage student attention.

    E:

    Spinner activity ( for calculating expected value, and odds calculation for choosing socks) requires students to record and enumerate outcomes, analyze results, and make conclusions.

    R:

    Individual problem solving for expected value outcomes of different basketball shots requires students to understand possible outcomes before the event and recording, analysis, and drawing conclusions after the event.

    E:

    Lesson 3 Exit ticket provides a record of student performance in computing probabilities.

    T:

    Group and partner work help students help each other. This lesson also includes communicating mathematical ideas with appropriate vocabulary, accurate note-taking to reinforce learning, and creating a useful resource (notes). The extension activity builds on learning with a more complex variation of an earlier problem.

    O:

    Game show simulation introduces and develops the concepts of odds and expected value. Varied experimental and theoretical models reinforce and demonstrate techniques to find outcomes and make predictions.

     

    IS.1 - Preparation
    Consider using graphic organizers (e.g., Frayer Model, Verbal Visual Word Association, Concept Circles) to review key vocabulary prior to the lesson.  
    IS.2 - All Students
    Consider preteaching the concepts critical to this lesson, including the use of hands-on materials. Throughout the lesson (based upon the results of formative assessment), consider the pacing to be flexible to the needs of the students. Also consider the need for reteaching and/or review both during and after the lesson as necessary.  
    IS.3 - Struggling Learners
    Consider providing struggling students with time to preview/review information on probability and odds at www.khanacademy.org  
    IS.4 - All Students
    Consider using the “Stick or Switch” activity  (see http://www.pdesas.org/module/content/resources/18082/view.ashx  found on PA’s SAS) to provide a concrete representation of expected value, probability and odds.  
    IS.5 - All Students
    Consider using a simulated virtual spinner (sample found on NCTM’s Illuminate : http://illuminations.nctm.org ).  
    IS.6 - Struggling Learners
    Consider providing struggling students with an example of the calculations for the expected value of this simulation.  
    IS.7 - All Students
    Consider using a concrete (see http://www.nctm.org/profdev/content.aspx?id=11688  ) way to show odds.  
    IS.8 - All Students
    Consider using the expected value activity  ( http://illuminations.nctm.org/LessonDetail.aspx?id=L761 ) which provides an opportunity to  student to gather the experimental data by interactively spinning as many trials as desired.  Students can also compare the results  of the theoretical and experimental data using the interactive activity.  
    IS.9 - All Students
    Consider providing students a copy of the definition of odds with detailed examples so that students can focus on calculating odds, not remembering the definition of odds.  
    IS.10 - All Students
    Consider allowing students the opportunity to complete this activity in pairs if a student hasn’t shown the proficiency to complete the work individually at this time.  
    IS.11 - All Students
    Also consider the use of Think-Alouds on the part of the students and the teacher.  
    IS.12 - All Students
    Consider planning this lesson while thinking about the common preconceptions and misconceptions that students will bring. This allows the instructor to plan strategic points for formative assessment strategies in the lesson and to plan answers to the questions that are likely to be asked. For additional information about questions that the instructor can ask the student (assessing and advancing questions), please read the Teachers’ Desk Reference: Essential Practices for Effective Mathematics Instruction by going to http://www.pattan.net/category/Resources/PaTTAN%20Publications/Browse/Single/?id=4e1f51d3150ba09c384e0000 
    IS.13 - All Students

    For additional information on formative assessment, please visit www.pdesas.org

    and click on the “Fair Assessments” tab

Instructional Procedures

  • View

    After this lesson, students will know how to convert from probability to odds and odds to probability. They will be able to calculate the expected value of an experiment and will understand that although a person may not necessarily achieve the expected value, in the long-run, the actual value will eventually reach the expected value.

    How many of you have seen or heard of the game show Wheel of Fortune[IS.4 - All Students] The wheel has 24 congruent sections in all and each has the same probability. There are different dollar values, prizes, or bankrupts on each. The people behind the show understand how to use expected value. They know the average amount of money someone will win. In probability, expected value is the sum of all the values multiplied by their probabilities.”

    Show students Spinner 1 (M-A2-1-3_Spinner 1.doc). [IS.5 - All Students]

    Each section has its own dollar value and unique probability of selection. The expected value is $1(1/2) + $2(1/4) + $3(1/8) + $4(1/16) + $5(1/16) = $1.9375. Of course a thousandth of a cent is not possible, but in the long run, the average amount of money someone would win on this spinner would be $1.9375.

    Here is another example. A fair number cube has 6 faces. Each face has a 1/6 probability of being rolled. The expected value of rolling a number cube is 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 3.5. Once again, it is not possible to roll a 3.5, but in the long run, if you were to keep track of all the rolls, the average roll would be 3.5.”

    Activity 1: Pairs

    Provide one spinner for each pair. Have students keep track of where they land for two spins, five spins, and ten spins. They should then compute the actual value they would win if this were a contest. [IS.6 - Struggling Learners]

    Another concept that is related to probability is ‘odds.’ If the probability of winning a contest is 1 out of 100 (or 0.01), then the odds of winning are 1:99. This is read ‘one to 99,’ meaning there is one way to win for every 99 ways to lose. In other words, the odds in favor of an outcome are ( p / 1 - p ) and odds against that outcome are ( 1 - p / p ).” [IS.7 - All Students]

    Examples:

    1. If the probability of losing is 5/16, what are the odds of winning?

    [11/16 ÷ ( 1 - 11/16 ) = 11/5 =11:5]

    1. If the odds of losing are 20:1, what is the probability of winning?

    [ p / 1 - p = 20/1, 21p = 20, p = 20/21, 1 - 20/21 = 1/21]

    Activity 2: Whole Class

    Distribute Spinner 2 (M-A2-1-3_Spinner 2.doc). Tell students to fill in the spinner with the following labels: three sectors should be “Bankrupt,” three sections should be “$1,” two sections should be “$2,” two sections should be “$5,” and the last six sections should be “$50,” “$25,” “$20,” “$10,” “$0.50,” and “$0.25.”

    1. Calculate the expected value.

    [3/16 • 0 + 3/16 • 1 + 2/16 • 2 + 2/16 • 5 + 1/16 • 50 + 1/16 • 25 + 1/16 • 20 + 1/16 • 10 +

    1/16 • 0.50 + 1/16 • 0.25 = $5.11]

    2. After two spins, what is the actual value? [Answers will vary.]

    3. What are the odds of winning $50? [probability = 1/16; odds 1:15]

    4. What are the odds of hitting bankrupt? [probability = 3/16; odds 3:13]

    Activity 3: Pairs

    There is a bag containing 5 colors of table tennis balls. Picking randomly, you have a 1/3 chance of getting a Red, a 1/4 chance of getting a Blue, a 1/4 chance of getting a Yellow, a 1/8 chance of getting a White, and a 1/24 chance of getting a Green. Each color of table tennis ball is labeled with a different number. Red is 4, Blue is 3, Yellow is 2, White is 1, Green is 0.

    1. What is the expected value? [1/3 • 4 + 1/4 • 3 + 1/4 • 2 + 1/8 • 1 + 1/24 • 0 = 2.7]

    2. What are the odds of picking a Green? [1/24 ÷ ( 1- 1/24 ) = 1/23 : 1:23]

    3. What is the probability of getting a Red or Blue or Yellow? [1/3 + 1/4 + 1/4 = 5/6]

    Activity 4: Think-Pair-Share

    Going back to the sock problem from the previous lesson, think about the following problems and then pair up and discuss your findings.” [IS.8 - All Students]

    Recap: There are 50 socks in a drawer; 20 are white, 12 are black, 10 are green, 6 are yellow, and 2 are red.

    1. What are the odds that you will randomly pick one yellow sock first? [ 6/50 = 0.12; 6/50 x 50/44 = 3/22 = 3:22 ]

    2. What are the odds that you will randomly pick one white or green sock first? [IS.9 - All Students]

    [ 20/50 + 10/50 = 30/50 = 0.6; 3/5 x 5/2 = 3/2 = 3:2 ]

    1. What are the odds that you will not randomly pick one red sock first?

    [ 1 − 2/50 = 48/50 = 0.96; 24/25 x 25/1 = 24/1 = 1:24 ]

    1. What are the odds that you will randomly pick one black and one white sock?

    [12/50 × 20/49 + 20/50 × 12/49 = 480/2450 = 0.196; 48/245 × 245/197 = 48/197 = 48:197]

    1. What are the odds that you will randomly pick a green sock last?

    [20/50 × 10/49 + 12/50 ×10/49 + 10/50 × 9/49 + 6/50 × 10/49 + 2/50 × 10/49 = 490/2450 = 0.2; 1/5 × 5/4 = 1/4 = 1:4]

    1. What are the odds that you will randomly pick two socks of the same color?

    [20/50 × 19/49 + 12/50 × 11/49 + 10/50 × 9/49 + 6/50 × 5/49 + 2/50 × 1/49 = 634/2450 = 0.2588; 317/1225 × 1225/908 = 317/908 = 317:908 ]

    Activity 5: Individual

    Use the following information to answer the following questions. [IS.10 - All Students] A basketball player is practicing shooting baskets in the gym alone. The probability that the player will make a layup is 0.99. The probability that the player will make a jump shot is 0.60. The probability that the player will make a basket from the foul line is 0.75. The probability that the player will make a 3-point basket is 0.45.

    1. If the player were keeping score, what is the expected value for each attempt? [Foul shot = 1 point, layup and jump shot = 2 points each, and 3-point basket = 3 points]

    2. What are the odds that the player will miss a foul shot? [1 − 0.75 = 0.25; 1/4 × 4/3 = 1/3 = 1:3 ]

    3. What are the odds that the player will make a 3-point basket and a foul shot?

    [0.45 × 0.75 = 0.3375; 27/80 × 80/53 = 27/53 = 27:53]

    4. What are the odds that the player will miss a layup?

    [1 − 0.99 = 0.01; 1/100 × 100/99 = 1/99 = 1:99]

    Distribute Lesson 3 Exit Ticket (M-A2-1-3_Lesson 3 Exit Ticket.doc and M-A2-1-3_Lesson 3 Exit Ticket KEY.doc) to evaluate students’ understanding.

    Extension:

    Use the following strategies to tailor the lesson to meet the needs of your students throughout the year.

    • Routine: Group and partner work is used throughout so that students can help each other. Emphasis should be placed on communicating mathematical ideas with the specific vocabulary words appropriate to the concepts. The lesson requires accurate note-taking skills to enhance the learning experience while creating a useful resource (notes).

    • Small Groups: Students needing opportunity for additional learning can be placed in one or more small groups to get further assistance from the instructor.

    • Expansion: Have students make a tree diagram of the basketball scenario from Activity 5. “Let’s say that the basketball player is only shooting four shots. What are all the possible outcomes and the probability of each?”

    For this lesson, the opening discussion grabs the students’ attention in that it pertains to something they are most likely familiar with: a game show. They are intrigued as to why there is a discussion about a game show in math class. It leads the teacher to introduce odds and expected value. Students are interested in games, especially if money is a prize. The topic is connected to activities from the earlier lessons and students understand how much depth these topics can have.

Related Instructional Videos

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DRAFT 05/28/2010
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