<p>This lesson will relate probability to the concept of odds. It will connect students' existing knowledge of probability and how to apply it to odds and "expected value." Students will:<br>- convert odds into probability and vice versa.<br>- determine the expected and actual values of an experiment.</p>

<p>- How are the probabilities of independent and dependent events determined, and what distinguishes them?</p>

<p>- Odds: The ratio of favorable outcomes to unfavorable outcomes in an event.&nbsp;<br>- 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.&nbsp;<br>- Independent Event: Two events in which the outcome of one event does not affect the outcome of the other.&nbsp;<br>- Dependent Event: Two events in which the outcome of one event affects the outcome of the other.&nbsp;<br>- 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.&nbsp;</p>

<p>- copies of Spinner 1 (M-A2-1-3_Spinner 1)<br>- number cubes<br>- copies of Spinner 2 (M-A2-1-3_Spinner 2)<br>- copies of Lesson 3 Exit Ticket (M-A2-1-3_Lesson 3 Exit Ticket and KEY)&nbsp;</p>

<p>- Teacher observations during group activities and class discussions provide the teacher with direct and helpful information about student engagement.&nbsp;<br>- The Lesson 3 Exit Ticket activity (M-A2-1-3_Lesson 3 Exit Ticket and KEY) keeps track of individual students' success in calculating and comparing probabilities.&nbsp;</p>

<p>Active Engagement and Explicit Instruction&nbsp;<br>W: Students learn to convert probability to odds and calculate the&nbsp;expected value for experimental or theoretical outcomes.&nbsp;<br>H: The game show simulation, Wheel of Fortune, engages students with a familiar and relevant situation.&nbsp;<br>E: The spinner task, which involves calculating expected value and probabilities for choosing socks, requires students to record and examine outcomes before drawing conclusions.&nbsp;<br>R: To solve individual problems for anticipated value outcomes of basketball shots, students must first grasp possible outcomes and then record, analyze, and draw conclusions.&nbsp;<br>E: The exit ticket for Lesson 3 records students' performance in computing probabilities.&nbsp;<br>T: Group and partner work promotes peer support among students. This lesson also covers conveying mathematical concepts using proper vocabulary, taking precise notes to encourage learning, and generating a useful resource (notes). The extension activity builds on previous learning by presenting a more complicated variation of an earlier challenge.&nbsp;<br>O: Game show simulation teaches the concepts of chances and expected value. Various experimental and theoretical models support and demonstrate approaches for determining outcomes and making predictions.&nbsp;<br>&nbsp;</p>

<p>After this lesson, students will understand how to convert from probability to odds and odds to probability. They will be able to determine the expected value of an experiment and comprehend that, while a person may not always achieve the expected value, the actual value will eventually equal the expected value.<br><br><strong>"How many of you have watched or heard of the game program </strong><i><strong>Wheel of Fortune</strong></i><strong>?&nbsp;The wheel has 24 congruent section in total, each with the same probability. There are various dollar values, awards, or bankruptcies on each. The show's creators understand how to use </strong><i><strong>expected value</strong></i><strong>. They understand the average amount of money that someone will win. In probability, the&nbsp;expected value equals the sum of all values multiplied by their probabilities."</strong><br><br>Show students Spinner 1 (M-A2-1-3_Spinner 1).&nbsp;<br><br><strong>"Each section has a different financial value and 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 conceivable, but over time, the average amount of money won on this spinner would be $1.9375.</strong><br><br><strong>Here's another example. A fair number cube has six faces. Each face has a 1/6 probability of being rolled. Rolling a number cube has an anticipated value of 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 3.5. Again, it is impossible to roll a 3.5, but if you keep track of all the rolls throughout time, the average would be 3.5."</strong><br><br><strong><u>Activity 1: Pairs</u></strong><br><br>Each pair will receive one spinner. Have students keep track of where they land after two, five, and ten rotations. They should next calculate the actual value they would receive if this were a competition.&nbsp;<br><br><strong>"The concept of 'odds' is related to probability. If the probability of winning a contest is one out of 100 (or 0.01), the odds of winning are 1:99. This is read 'One to 99', indicating that there is a single possibility of winning for every 99 possibilities of losing. In other words, the odds in favor of a result are (p / 1 - p), while the odds against that outcome are (1 - p / p)."&nbsp;</strong><br><br>Examples:<br><br>1. What are the probability of winning if the probability of losing is 5/16?<br><br>[11/16 ÷ ( 1 - 11/16 ) = 11/5 = 11:5]<br><br>2. What is the probability of winning if the odds of losing are 20:1?<br><br>[ p / 1 - p = 20/1, 21p = 20, p = 20/21, 1 - 20/21 = 1/21]<br><br><strong><u>Activity 2: Whole Class</u></strong><br><br>Distribute Spinner 2 (M-A2-1-3_Spinner 2). Tell students to fill out 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 final six sections should be "$50," "$25," "$20," "$10," "$0.50," and "$0.25".<br><br>1. Determine the expected value.<br><br>[<i>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</i>]<br><br>2. What is the true value after two spins? [<i>Answers will vary.</i>]<br><br>3. What are the odds of winning 50 dollars? [<i>probability = 1/16; odds 1:15</i>]<br><br>4. What are the odds of going bankrupt? [<i>probability = 3/16; odds 3:13</i>]<br><br><strong><u>Activity 3: Pairs</u></strong><br><br>There is a bag with 5 different colors of table tennis balls. Picking at random gives you a 1/3 chance of receiving a Red, 1/4 chance of getting a Blue, 1/4 chance of getting a Yellow, 1/8 chance of getting a White, and 1/24 chance of getting a Green. Each color of a&nbsp;table tennis ball is identified by a separate number. Red is 4, blue is 3, yellow is 2, white is 1, and green is 0.<br><br>1. What is the expected result? [<i>1/3 • 4 + 1/4 • 3 + 1/4 • 2 + 1/8 • 1 + 1/24 • 0 = 2.7</i>]<br><br>2. What are the odds of choosing a Green? [<i>1/24 ÷ ( 1- 1/24 ) = 1/23 : 1:23</i>]<br><br>3. What is the probability of getting a Red or Blue or Yellow? [<i>1/3 + 1/4 + 1/4 = 5/6</i>]<br><br><strong><u>Activity 4: Think-Pair-Share</u></strong><br><br><strong>"Consider the following difficulties in relation to the sock problem from the previous lesson. Then, work in pairs to explain your findings."&nbsp;</strong><br><br><strong>Recap:</strong> A drawer contains 50 socks: 20 white, 12 black, 10 green, 6 yellow, and 2 red.<br><br>1. What are the odds of randomly selecting one yellow sock first? [<i>6/50 = 0.12; 6/50 × 50/44 = 3/22 = 3:22</i>]<br><br>2. What are the odds of randomly selecting one white or green sock first?&nbsp;<br><br>[<i>20/50 + 10/50 = 30/50 = 0.6; 3/5 x 5/2 = 3/2 = 3:2</i>]<br><br>3. What are the odds that you will not randomly select one red sock first?<br><br>[<i>1 − 2/50 = 48/50 = 0.96; 24/25 × 25/1 = 24/1 = 1:24</i>]<br><br>4. What are the odds of randomly selecting one black and one white sock?<br><br>[<i>12/50 × 20/49 + 20/50 × 12/49 = 480/2450 = 0.196; 48/245 × 245/197 = 48/197 = 48:197</i>]<br><br>5. What are the odds of randomly selecting a green sock last?<br><br>[<i>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</i>]<br><br>6. What are the odds of randomly selecting two socks of the same color?<br><br>[<i>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</i>]<br><br><br><strong><u>Activity 5: Individual</u></strong><br><br>Use the information provided to answer the following questions. A basketball player is practicing shooting baskets in the gym by himself. The probability of the guy making a layup is 0.99. The probability of the player making a jump shot is 0.60. The probability of the player making a basket from the foul line is 0.75. The player has a 0.45 probability of making a three-pointer basket.<br><br>1. What would be the expected value for each attempt if the player kept score? [<i>Foul shot = 1 point, layup and jump shot = 2 points each, and 3-point basket = 3 points</i>]<br><br>2. What is the odds that the player will miss a foul shot? [<i>1 − 0.75 = 0.25; 1/4 × 4/3 = 1/3 = 1:3</i>]<br><br>3. What are the odds that the player will make a 3-pointer and a foul shot?<br><br>[<i>0.45 × 0.75 = 0.3375; 27/80 × 80/53 = 27/53 = 27:53</i>]<br><br>4. What is the odds that the player will miss a layup?<br><br>[<i>1 − 0.99 = 0.01; 1/100 × 100/99 = 1/99 = 1:99</i>]<br><br>To assess students' understanding, distribute Lesson 3 Exit Tickets (M-A2-1-3_Lesson 3 Exit Ticket and KEY).<br><br><strong>Extension:</strong><br><br>Use the strategies listed below to modify the lesson to your student's needs throughout the year.<br><br><strong>Routine: </strong>Students aid one another through group and partner work. The emphasis should be on explaining mathematical ideas using vocabulary phrases&nbsp;specific to the subject. The class requires accurate note-taking skills in order to improve the learning experience while also providing a helpful resource.<br><br><strong>Small Groups:</strong> Students who require further learning might be assigned to one or more small groups and receive additional support from the instructor.<br><br><strong>Expansion: </strong>Ask students to create a tree diagram of the basketball situation from Activity 5. <strong>"Let's imagine the basketball player only takes four shots. What are all the possible outcomes, and what is the probability of each?"</strong><br><br>The opening discussion of this lesson is likely to captivate the students due to its relevance to a game show that they are already familiar with. They're curious why there's a discussion about a game show in math class. It prompts the teacher to discuss odds and expected values. Students are engaged in games, especially when there is a monetary award. The topic is connected to exercises from previous lessons, and students learn the depth that these themes can have.</p>