<p>This lesson expands on students' existing knowledge of probability. Students will:<br>- understand the various definitions associated with compound probability.<br>- calculate several types of probability, including intersections and unions.<br>- calculate the theoretical and experimental probabilities.</p>

<p>- How may data be arranged and portrayed so that the link between quantities is clear?<br>- How can we utilize probability and data analysis to make predictions?<br>- How does the type of data affect the display method?<br>- How precise should measurements and calculations be?<br>- How are mathematical properties of things or processes measured, calculated, and/or interpreted?<br>- What factors determine whether a tool or method is appropriate for a specific task?<br>- How are the probabilities of independent and dependent events determined, and what distinguishes them?</p>

<p>- Compound Event: An event made up of two or more simple events.&nbsp;<br>- Independent Event: Two events in which the outcome of one event does not affect the outcome of the other.&nbsp;<br>- Event: An outcome whose probability can be obtained from a single occurrence.&nbsp;<br>- Experimental Probability: A statement of probability based on the results of a series of trials.&nbsp;<br>- Theoretical Probability: A statement of the probability of an event without doing an experiment or analyzing data. - 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;<br>- Dependent Event: Two events in which the outcome of one event affects the outcome of the other.&nbsp;<br>- Complement: The negation of the occurrence of an event; the complement of event A is event A not occurring.&nbsp;<br>- Factorial: The product of all the integers less than or equal to the given integer; for example, the factorial of 5 is 5 × 4 × 3 × 2 × 1 and is expressed as 5!&nbsp;<br>- Permutation: An ordered arrangement of all or part of a set of objects.&nbsp;<br>- Combination: Any selection of one or more members of a set of objects without regard to order.</p>

<p>- cardboard boxes labeled <i>1</i>, <i>2</i>, and <i>3</i><br>- either two pictures of a goat or two stuffed-animal goats<br>- prizes such as candy, a notebook, markers, pens<br>- copies of Spinner Activity Worksheet (M-A2-1-1_Spinner Activity Worksheet)<br>- decks of cards (1 per group of 4 students)<br>- copies of Lesson 1 Exit Ticket (M-A2-1-1_Lesson 1 Exit Ticket and KEY)</p>

<p>- The Think-Pair-Share activity helps students evaluate each other's work.&nbsp;<br>- Teacher observations during group activities and class discussions provide the teacher with instant and helpful information on student engagement.<br>- The Exit Ticket activity (M-A2-1-1_Lesson 1 Exit Ticket and KEY) provides the teacher with a record of student achievement.&nbsp;<br>&nbsp;</p>

<p>Active Engagement, Modeling&nbsp;<br>W: Students develop an understanding of the distinctions between independent and dependent events, theoretical and experimental probabilities, and the ability to effectively utilize the tools to calculate compound and conditional probabilities.<br>H: Let's Make a Deal, a game show, serves as an introduction to the concept of compound probability.<br>E: Vocabulary references (e.g. event, independent, dependent, conditional, theoretical, experimental, Fundamental Counting Principle, factorial permutation, combination) from <a href="http://mathworld.wolfram.com/"><span style="color:#1155cc;"><u>http://mathworld.wolfram.com</u></span></a> connect probability concepts to student experiences.<br>R: Group activity: Card deck probability questions from random cards focus students' attention on outcomes that can be assessed and recorded.<br>E: Lesson 1&nbsp;Exit Ticket assesses students' grasp of the computations that underpin theoretical probability and provides a framework for comparing results.<br>T: Group and partner work, specialized vocabulary words, and note-taking abilities increase learning and serve as a valuable resource.<br>O: The lesson starts with a gaming exercise. Vocabulary connects significant words and concepts. Activities and questions allow students to explore probability in an open-ended way. Review and evaluation demonstrate what students have learned and anticipate the following topics in probability.</p>

<p>After this lesson, students will comprehend the distinction between independent and dependent events, as well as theoretical and experimental probability. They will be able to calculate both compound and conditional probabilities. Students learn best when they find a reason for learning the content. Probability occurs all around us, and when the examples are circumstances to which students can connect, their motivation to study and their success in the unit concepts will improve.<br><br>Introduce the game show "Let's Make a Deal" to the students.&nbsp;<br><br><strong>"Suppose you're on a game show and you're given the option of three doors: one leads to a car, the others to goats. You select a door, say No. 1, and the host, who knows what is behind the doors, opens another door, say No. 3, which contains a goat. He then asks, 'Do you want to pick door No. 2?' Is it advantageous for you to change your mind?"</strong><br><br>Conduct a few game demonstrations with students acting as contestants. Use cardboard boxes, one containing a gift and the other two with two stuffed animal goats (or goat-related photographs). After each participant has chosen, ensure that the contents of each box are mixed up. Tell students who are watching to take notes and calculate the chances of winning the prize.<br><br>After the demonstration, discuss the probability of winning. Show the graph that compares individual outcomes with and without switching.<br><br><strong>"After selecting a box, the host opens another and offers you the option to swap boxes. What are the chances you'll win the prize?"</strong> The frequent misperception about this problem is that the chance is 1/2, and most students will answer 1/2. The incorrect assumption is that there are only two remaining doors, while in fact there are three, and the probability is 1/3.<br><br>Show the Monty Hall Problem video on YouTube: <a href="https://www.youtube.com/watch?v=mhlc7peGlGg"><span style="color:#1155cc;"><u>https://www.youtube.com/watch?v=mhlc7peGlGg</u></span></a>.<br><br>Talk about their responses to the video and the problem. They can also see a demonstration at <a href="http://demonstrations.wolfram.com/MontyHallProblem/"><span style="color:#1155cc;"><u>http://demonstrations.wolfram.com/MontyHallProblem/</u></span></a>.<br><br>Students should record the following vocabulary in their notebooks.&nbsp;The vocabulary is from <a href="http://mathworld.wolfram.com/"><span style="color:#1155cc;"><u>http://mathworld.wolfram.com</u></span></a>.<br><br>An <strong>event</strong> is the result of an experiment. For example, flipping a coin produces two outcomes: heads or tails.&nbsp;<br><br><strong>P(A)</strong> represents the probability of event A occurring.<br><br><strong>1 − P(A)</strong> represents the chance of event A <i>not occurring</i>.<br><br><strong>P(A ∩ B)</strong> represents the chance that events A <i>and</i> B will occur.<br><br><strong>P(A ∩ B) = P(A) × P(B).</strong><br><br><strong>P(A U B)</strong> represents the probability of event A <i>or</i> event B occurring.<br><br><strong>P(A U B) = P(A) + P(B) − P(A ∩ B)</strong><br><br>One event occurring does not change the probability of the other event, making the two events <strong>independent.</strong> For example, if you flip a coin twice, getting heads on the first flip has no bearing on the probability of getting tails on the second. It is still 1/2.<br><br>This indicates that if two occurrences are <strong>dependent</strong>, the occurrence of one influences the probability of the other. For example, the probability of selecting a red card from a deck of 52 cards is 26/52 = 0.5 (since half of the cards in the deck are red). If you keep that card, the probability of getting another red card is now 25/51 = 0.49, because the card you've removed is one of the red cards (26 - 1) and one of the total 52 cards (52 - 1).<br><br>The Monty Hall problem is an illustration of <strong>conditional probability</strong>. Conditional probability is the probability of one event occurring given that another has already occurred.<br><br><strong>P(A|B) </strong>represents the probability of event A occurring if event B has already happened: <strong>P(A|B) = P(A ∩ B) ÷ P(B).</strong><br><br><strong>Theoretical probability</strong> is the expected outcome, whereas <strong>experimental probability</strong> is the actual outcome. For instance, since the number 4 appears on one of the six sides of the number cube, the theoretical probability of rolling a 4 is 1/6 = 0.167. If you ran an experiment and rolled a 4 once out of ten rolls, the experimental probability is 1/10 = 0.10.<br><br>The <strong>Fundamental Counting Principle</strong> is a mechanism for calculating all possible outcomes of a particular event. For instance, if you have three jeans and five shirts to go to school, you can create 15 alternative ensembles.<br><br>A <strong>factorial</strong> is the product of an integer and all integers less than or higher than zero. It is denoted as "<strong>n!</strong>." For example, 4! = 4 × 3 × 2 × 1 = 24. Notice that there is only one method to arrange zero objects, and the expression as a factorial is 0! = 1.<br><br>A <strong>permutation</strong> is the number of ways to arrange items in order. For example, if Alex, Ben, and Colleen are competing for student council president, vice president , and treasurer, there are six possible arrangements: ABC, ACB, BAC, BCA, CAB, and CBA. Consider the p in permutation to indicate position.<br><br>A <strong>combination</strong> is the number of possible arrangements in which order is irrelevant. For example, if Alex, Ben, and Colleen are competing for two spots on a sports team, they can only select one of three choices: AB, AC, or BC. Making the team with Alex and Ben is identical to making the team with Ben and Alex (order is irrelevant). Consider the c in combination as a symbol for the&nbsp;committee.<br><br><strong><u>Activity 1: Pairs</u></strong><br><br>If possible, place students in a computer lab or have them use laptops. If this is not possible, you can complete the exercise as a class. Distribute the Spinner Activity Worksheet (M-A2-1-1_Spinner Activity Worksheet).<br><br>Navigate to <a href="http://illuminations.nctm.org/ActivityDetail.aspx?id=79"><span style="color:#1155cc;"><u>http://illuminations.nctm.org/ActivityDetail.aspx?id=79</u></span></a>. It has a spinner on it that can be changed to indicate the number of sectors. Instead of utilizing the data provided on the website, have students calculate the theoretical and experimental probabilities.<br><br><strong><u>Activity 2: The Birthday Problem</u></strong><br><br>Ask students to consider the likelihood of any two classmates in class having birthdays on the same day of the week (e.g., Monday, or&nbsp;Tuesday).<br><br><strong>"The probability of two students having birthdays on the same day of the week is (7/7) × (1/7) = .14285. You begin with 7/7 because the first student's birthday can occur on any day of the week, and you multiply by 1/7 because the second student's birthday must fall on the same day of the week. To calculate the total probability, multiply the probabilities of both events together. Working with the chance of an event not occurring can be easier to calculate than the probability of an event occurring. The probability that two people do not share the same birthday is (365/365) × (364/365) = 0.99726. The first student's birthday has a probability of 365/365 because it can occur on any day of the year. The second student's birthday has a probability of 364/365 because it can occur on any day other than one. Thus, the likelihood of two people sharing the same birthday is 1 - 0.99726 = 0.00274."</strong><br><br>Visit <a href="http://demonstrations.wolfram.com/TheBirthdayProblem/"><span style="color:#1155cc;"><u>http://demonstrations.wolfram.com/TheBirthdayProblem/</u></span></a>. Show students the demonstration.&nbsp;Change the number of people and students in the classroom. This provides them with&nbsp;a theoretical probability. Then ask students to compute the experimental probability.<br><br><strong><u>Activity 3: Think-Pair-Share</u></strong><br><br>With students, discuss <i>sample size</i>. In the problem below, the sample size begins at 20 candies. Each time a candy is taken, the sample size goes down by one unless it is refilled. The concept of replacement is vital, and it should be stressed by an experiment. A bag contains a blue and a scarlet marble. The probability of selecting the blue marble at random is&nbsp;half. If the blue marble is <u>not</u> replaced in the next random draw, the probability of drawing the red marble is 1, while the probability of drawing the blue marble is 0. However, if you change the blue marble after the initial draw, the probability remains 0.5. Point out that these exercises differ from coin flips or rolls of one or more number cubes in that they do not involve replacement. Also, clarify the distinction between a flip and a roll. A coin flip has two outcomes, but a roll has six.<br><br>Present the following scenario to students:<strong> "You're sitting in front of the principal's desk when you notice she has a jar of candy.&nbsp;You have 10 strawberry-flavored candies, 5 vanilla-flavored, 4 blueberry-flavored, and 1 chocolate-flavored in the jar."</strong><br><br>1. What is the probability of each flavor?<br><br><i>P(S) = 10/20 &nbsp;&nbsp;&nbsp;&nbsp; P(V) = 5/20</i><br><br><i>P(B) = 4/20 &nbsp;&nbsp;&nbsp; &nbsp; P(C) = 1/20</i><br><br>2. The principal informs you that you may only select one candy at random. What is the probability of picking blueberries or strawberries? <i>P(B U S) =</i><br><br><i>P(B) + P(S) = (4/20 + 10/20 ) = 14/20</i><br><br>3a. She tells you that you can choose two candies at random. After picking the first candy, place it back in the jar. What is the probability of selecting one vanilla and one chocolate?<br><br><i>P(V ∩ C) =( 5/20 × 1/20 ) = 5/400 = 1/80 = 0.0125</i><br><br>3b. If the first candy is not returned to the jar, what is the probability of selecting one vanilla and one chocolate? (Remember that the second candy reduces the sample size, or denominator, by one. The first product is the probability of selecting vanilla first and then chocolate. The second product is the probability of pulling a chocolate first, followed by a vanilla.)<br><br>( 5/20 × 1/19 ) + ( 1/20 × 5/19 ) = 5/380 + 5/380 = 10/380 = 0.02632<br><br>4a. What is the probability of randomly selecting one blueberry after selecting and replacing a vanilla candy in the jar?<br><br><i>P(B|V) = (4 blueberry/20 candies ) = (0.2)</i><br><br>4b. What is the probability of randomly selecting one blueberry if you have already chosen a vanilla candy and have not changed it?<br><br><i>P(B|V) = (4 blueberry/19 candies ) =&nbsp; (4/19).</i><br><br>5. Assume that each candy in the jar has a number on it. Numbers 1 through 10 were assigned to the ten strawberry candies, 1 through 5 to the five vanilla sweets, 1 through 4 to the four blueberry candies, and 1 to the chocolate. Using the basic counting principle, how many ways can you choose one of each color? <i>(200)</i><br><br><strong><u>Activity 4: Groups</u></strong><br><br>Divide the students into groups of four.&nbsp;Hand them each a deck of cards. Give them 10 minutes to create ten probability questions on drawing from the deck. For example, what is the probability of selecting one red card? What is the probability of selecting a red card <i>and</i> a four? After they've written their questions, have each group share one with the entire class and write it on the board. When there are ten questions on the board, the groups must solve them. Review and debate the answers as a class. Collect each group's questions for use as a warm-up the next day or on a quiz.<br><br>An Exit Ticket (M-A2-1-1_Lesson 1 Exit Ticket and KEY) is a quick way for determining whether students understand the concepts.<br><br><strong>Extension:</strong><br><br>Use the technique below to adjust 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 placed on communicating mathematical ideas using vocabulary words&nbsp;appropriated to the topics. The class requires accurate note-taking skills in order to improve the learning experience while also providing a helpful resource.<br><br>This lesson serves as a foundation for subsequent classes and is required for students to comprehend probability and its application in real-world situations. It starts with a game show exercise to captivate and spark students' interest. The teacher then teaches some language that the students will need to know before to the activities. The tasks are easy, giving students time to investigate probability. Students are given time to review the class content and receive timely feedback. The teacher also outlines how students will apply this knowledge in the upcoming class.</p>