<p>This lesson will connect the vocabulary from the first lesson to a visual representation of probability problems. Students will:<br>- be able to organize the information provided into a tree diagram.<br>- be able to compute compound probabilities using a tree diagram.</p>

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

<p>- Tree diagram: A graphic illustration that shows all possible outcomes for a process with one or more stages.&nbsp;<br>- Replacement: In probability experiments, the reintroduction of the selected object to the total population, following the removal of the selected object in accordance with the experiment.&nbsp;<br>- Probability: The mathematical probability of an event is represented by a real number, <i>p</i>, such that , where an impossible event is 0 and a certain event is 1.&nbsp;<br>- Complement: The probability of an event not occurring. If the probability of the occurrence of an event is <i>p</i>, then its complement is 1 – <i>p</i>.&nbsp;<br>- Sample Space: The systematic representation, by table, chart, diagram, or list of all possible outcomes of an event.&nbsp;<br>- Fundamental Counting Principle: If there are a ways to do one thing and b ways to do another, then there are <i>a • b</i> ways to do both things.</p>

<p>- big clear bowl<br>- table tennis balls with numbers written on them (1 through however many students are in class)<br>- poster paper<br>- markers<br>- copies of Lesson 2 Exit Ticket (M-A2-1-2_Lesson 2 Exit Ticket and KEY)&nbsp;</p>

<p>- Think-pair-share and group work observations help students&nbsp;assess one another's work.&nbsp;<br>- Teacher observation during class discussions and activities provides the teacher with promptly helpful information about student engagement.<br>- Lesson 2 Exit Ticket activity (M-A2-1-2_Lesson 2 Exit Ticket and KEY) keeps track of individual student abilities in deciding outcomes.&nbsp;<br>&nbsp;</p>

<p>Active Engagement, Modeling&nbsp;<br>W: Students learn to design and analyze probability trees/diagrams, understand their relationship to compound probabilities, and know when to multiply or add probabilities.&nbsp;<br>H: Randomly selecting from candy jars with different flavors helps kids focus on the possibility of each event.&nbsp;<br>E: Pair-share&nbsp;activities, such as predicting basketball foul shots, ice cream scoop combinations, and class presentation orders, allow students to learn about theoretical probability and its applications.&nbsp;<br>R: A pair activity on randomly selecting socks asks students to record, enumerate, and analyze outcomes to compare probabilities.&nbsp;<br>E: The lesson 2 exit ticket evaluates students' proficiency in probability computation.&nbsp;<br>T: The lesson focuses on group and partner work, effective communication of mathematical concepts, note-taking for reinforcement, and developing a useful resource. Students who require further learning may be put in one or more small groups to receive further support from the instructor.&nbsp;<br>O: The lesson engages students with real-world scenarios, games that require knowledge of outcomes, and pair and group activities that encourage student explanations.</p>

<p>Students will be able to organize information into a probability tree or tree diagram after completing this lesson. They will grasp how to read tree diagrams and how they are related to compound probabilities. They will grasp when to multiply and add probabilities.<br><br><strong>"Recall the problem with the candy jar on the principal's desk? Today we will learn how to organize that information into a tree diagram. Remember that each jar included 10 strawberry sweets, 5 vanilla, 4 blueberry, and 1 chocolate. What is the probability of each flavor?" </strong>[<i>10/20 = 1/2, 5/20 = 1/4, 4/20 = 1/5, 1/20</i>]<br><br><strong>"What is the total probability?" </strong>(<i>1</i>)<br><br>(Note: What you pick in the first jar has no effect on what you pick in the second jar.)<br><br>Students should record the following tree diagram in their notes.<br><br><strong>"This is an illustration of a tree diagram. Each branch indicates one or more possible events, as well as their probability. The branches add up to a probability of 1. The first four branches indicate picking a candy from the jar, while the second set represents selecting another sweet from the jar. The events and probability on the right show how the two branches intersect. To calculate the probability of receiving at least one strawberry, sum all events with the letter "S."</strong> [<i>.25 + .125 + .10 + .025 + .125 + .10 + .025 = .775</i>]</p><figure class="image"><img style="aspect-ratio:764/661;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_34.png" width="764" height="661"></figure><p><strong>"This tree diagram depicts two independent events. But what if the principle allowed you to choose two candies? What is the probability of picking a strawberry on the second pick if you choose one first?"</strong><br><br>(<i>You picked a strawberry first, therefore there are now 9 strawberries remaining, and the total is now 19, or 9/19 = .474</i>)<br><br><strong>"What is the probability of picking 2 strawberry candies from one jar?"</strong><br><br>[<i>10/20 × 9/19 = .50 × .474 = .237</i>]<br><br><strong>"This 'without replacement' situation implies that the second choice is dependent on the first. The results of the second selection would have been independent of the first if this was a "with replacement" situation and you had selected a candy and subsequently returned it."</strong><br><br><strong>According to the Fundamental Counting Principle, there are </strong><i><strong>a • b</strong></i><strong> ways to do both tasks if there are </strong><i><strong>a</strong></i><strong>&nbsp;ways to complete one task and </strong><i><strong>b</strong></i><strong> ways to complete another. If someone owns 3 different color t-shirts and 2 different color shorts, they can choose one t-shirt and one pair of shorts from 3 × 2 = 6 combinations.</strong><br><br><strong><u>Activity 1: Think-Pair-Share</u></strong><br><br>Write the following problem on the board or give it to students to think about on their own, and then pair them up to answer the questions. When everyone has finished the problem, have the pairs present their findings to the class.<br><br><strong>"In basketball, a player gets to shoot two free-throws if he or she is fouled while shooting and the shot does not go in the basket. Melanie makes her free-throws 6 times out of 10 attempts. The sample space lists all possible event outcomes."</strong><br><br>1. What are all the possible results of her two shots?<br><br>[<i>(make, make); (make, miss); (miss, make); (miss, miss) These are the outcomes that make up the sample space</i>.]<br><br>2. Find the following probabilities:<br><br>a. P(make, make) = (<i>0.6 × 0.6 = 0.36</i>)<br><br>b. P(make, miss) = (<i>0.6 × 0.4 = 0.24</i>)<br><br>c. P(miss, make) = (<i>0.4 × 0.6 = 0.24</i>)<br><br>d. P(miss, miss) = (<i>0.4 × 0.4 = 0.16</i>)<br><br>3. What is the probability she will make at least one shot?<br><br><i>(0.36 + 0.24 + 0.24 = 0.84)</i><br><br>4. How likely are her two shots to be the same—that is, make-make or miss-miss? (<i>0.36 + 0.16 = 0.52</i>)<br><br><strong><u>Activity 2: Groups</u></strong><br><br>Assign the following problem to each of the four groups in class to complete.<br><br><br><strong>"You're at an ice cream shop that only serves the two most popular flavors: chocolate and vanilla. You may use as many scoops of each flavor as you like per serving. You select chocolate on eight out of ten visits."</strong><br><br>1. List every option you have when selecting a three-scoop ice cream cone.<br><i>CCC &nbsp;CCV &nbsp;CVC &nbsp;VCC</i><br><i>VVV &nbsp;VVC &nbsp;VCV &nbsp;CVV</i><br><br>2. What is the probability that you get a cone with at least one scoop of vanilla? [<i>7/8 = 0.875</i>]<br><br>3. What is the probability that you get a cone with exactly two scoops of chocolate? [<i>3/8 = 0.375</i>]<br><br>4. What is the probability that you get a cone with all the same flavor? [<i>2/8 = 0.25</i>]<br><br>5. What is the probability that you get a scoop of chocolate on the third scoop, given you got a scoop of vanilla on the first scoop? [<i>2/8 = 0.25</i>]<br><br>6. One day per month, the ice cream parlor sells another flavor. You decide that you want a three scoop cone and that you do not want to repeat flavors. How many possible ice cream cones can you get? (Hint: order matters) [<i>3! = 3 × 2 × 1 = 6</i>]<br><br>7. Write two of your own questions for this scenario.<br><br><strong><u>Activity 3: Groups</u></strong><br><br>For this task, divide the students into groups of four. Give them the following task to solve, and walk around to answer any questions they may have. Use a large clear bowl filled with table tennis balls as a visual.<br><br><strong>"Assume that your Language Arts class requires your group of four to do two presentations. Each day, only two groups are present. Your teacher assigns the sequence of presentations using a lottery system. Each group is given a number (1–4). Because you have to give two presentations, the teacher will pick a ball at random and then replace it. So it's possible that you'll have to do both presentations on the same day. Every ball has the same probability. Use this information to respond to the following questions."</strong><br><br>1. How many possible results are there? [<i>16</i>]&nbsp;<br><br><br>2. Write out all of the possible outcomes for this situation. (<i>Note: (1, 1) indicates that the group was picked to go first for presentation #1 and then for presentation #2.)</i><br><br><i>(1, 1) (2, 2) (3, 3) (4, 4) (1, 2) (1, 3) (1, 4) (2, 1)</i><br><br><i>(2, 3) (2, 4) (3, 1) (3, 2) (3, 4) (4, 1) (4, 2) (4, 3)</i><br><br>3. What is the probability of every possible result? [<i>0.625 or 1/16</i>]<br><br>4. What is the probability that group 3 will initiate action first? [<i>0.45 or 0.25</i>]<br><br>5. What is the probability that on day one, group 4 will go at least once?<br><br>[<i>1/4 + 1/16 + 1/16 + 1/16 + 1/16 = 1/2</i>]<br><br>6. On the first day, what is the probability that one group will go twice?<br><br>[<i>1/16 + 1/16 +1/16 = 3/16</i>]<br><br><strong><u>Activity 4: Pairs</u></strong><br><br>Although it uses a tree diagram, this problem includes "without replacement." Students should work in pairs to tackle the following issue.&nbsp;<br><br><br>Your sock drawer is disorganized as you get ready for school. You have 20 white socks, 12 black, 10 green, 6 yellow, and 2 red.<br><br>1. Make a tree diagram for this scenario on&nbsp;poster paper. Take special note of how this scenario differs from the other tree diagrams we have created. [Hint: after removing a sock, it should never be replaced in the drawer.]</p><figure class="image"><img style="aspect-ratio:928/882;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_35.png" width="928" height="882"></figure><p>2. What are all the potential results, and what is the probability of each one?&nbsp;<br><br>3. What is the probability of selecting a single black sock?&nbsp;<br><br>[<i>0.0980 + 0.0980 + 0.0539 + 0.0490 + 0.0294 + 0.0098 + 0.0490 + 0.0294 + 0.0098 = 0.42634</i>]&nbsp;<br><br>4. What is the probability of selecting one red sock or one yellow sock?&nbsp;<br><br>(<i>0.48, excludes one red and one yellow</i>)&nbsp;<br><br>5. What is the probability that the two pairs of socks have the same color?&nbsp;<br><br>[<i>0.1551 + 0.0539 + 0.0367 + 0.0122 + 0.0008 = 0.02587</i>]&nbsp;<br><br>To assess students' comprehension, distribute the Lesson 2 Exit Ticket (M-A2-1-2_Lesson 2 Exit Ticket and KEY).<br><br><strong>Extension:</strong><br><br>Use the strategies listed below to adjust the lesson to your students' 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 that are specific to the subject. The class requires accurate note-taking skills to improve the learning experience and create a useful resource.<br><br><strong>Small Groups:</strong> Students who require more learning opportunities might be assigned to one or more small groups and receive additional assistance from the instructor.<br><br>This lesson begins with a problem that relates to the previous lesson. Probability is an easy topic to connect to real-world circumstances that students can identify with. They enjoy playing games and understanding what their odds are in specific situations. This lesson includes both pair and group work since some students find these concepts challenging, and students often explain them better to one another.</p>