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:
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If the probability of losing is 5/16, what are the odds of winning?
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[11/16 ÷ ( 1 - 11/16 ) = 11/5 =11:5]
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If the odds of losing are 20:1, what is the probability of winning?
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[ 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.”
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Calculate the expected value.
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[3/16 • 0 + 3/16 • 1 + 2/16 • 2 + 2/16 • 5 + 1/16 • 50 + 1/16 • 25 + 1/16 • 20 + 1/16 • 10 +
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1/16 • 0.50 + 1/16 • 0.25 = $5.11]
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2. After two spins, what is the actual value? [Answers will vary.]
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3. What are the odds of winning $50? [probability = 1/16; odds 1:15]
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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.
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1. What is the expected value? [1/3 • 4 + 1/4 • 3 + 1/4 • 2 + 1/8 • 1 + 1/24 • 0 = 2.7]
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2. What are the odds of picking a Green? [1/24 ÷ ( 1- 1/24 ) = 1/23 : 1:23]
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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.
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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 ]
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2. What are the odds that you will randomly pick one white or green sock first? [IS.9 - All Students]
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[ 20/50 + 10/50 = 30/50 = 0.6; 3/5 x 5/2 = 3/2 = 3:2 ]
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What are the odds that you will not randomly pick one red sock first?
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[ 1 − 2/50 = 48/50 = 0.96; 24/25 x 25/1 = 24/1 = 1:24 ]
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What are the odds that you will randomly pick one black and one white sock?
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[12/50 × 20/49 + 20/50 × 12/49 = 480/2450 = 0.196; 48/245 × 245/197 = 48/197 = 48:197]
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What are the odds that you will randomly pick a green sock last?
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[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]
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What are the odds that you will randomly pick two socks of the same color?
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[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 ]
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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.
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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]
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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 ]
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3. What are the odds that the player will make a 3-point basket and a foul shot?
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[0.45 × 0.75 = 0.3375; 27/80 × 80/53 = 27/53 = 27:53]
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4. What are the odds that the player will miss a layup?
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[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.
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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).
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Small Groups: Students needing opportunity for additional learning can be placed in one or more small groups to get further assistance from the instructor.
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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.
