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Equally Likely and Unequally Likely Outcomes

Lesson Plan

Equally Likely and Unequally Likely Outcomes

Objectives

This lesson reviews methods of finding experimental probabilities and the calculation of theoretical probabilities. Students will develop an understanding of the distinction between equally likely and unequally likely events and their associated probabilities. Students will:

  • compare experimental and theoretical probability.
  • use experimental probability to make predictions and conjectures.
  • understand the distinction between equally likely and unequally likely events.
  • write a definition for equally likely outcomes and be able to give examples of simple experiments that have equally likely outcomes.

Essential Questions

  • In what ways are the mathematical attributes of objects or processes measured, calculated, and/or interpreted?
  • How can probability and data analysis be used to make predictions?

Vocabulary

  • Complementary Event: The opposite of an event. That is, the set of all outcomes of an experiment that are not included in an event.
  • Equally Likely: Two or more possible outcomes of a given situation that have the same probability. If you flip a coin, the two outcomes—the coin lands heads up and the coin lands tails up—are equally likely to occur.
  • Outcome: One of the possible events in a probability situation.
  • Probability: A number from 0 to 1 that indicates how likely something is to happen.

Duration

90–120 minutes

Prerequisite Skills

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

Materials

Related Unit and Lesson Plans

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

http://www.random.org

  • Sophisticated and flexible random-number generator and number-cube roller

http://roll-dice-online.com

Formative Assessment

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    • Review student vocabulary journals for new entries and evaluate the definitions for accuracy.
    • Observe student performance during class discussion and small-group work to identify which students might need additional help and which concepts might need additional work.
    • Use the Partner Quick Quiz (M-7-2-1_Partner Quick Quiz and KEY.doc) to assess student understanding of the probability concepts.

Suggested Instructional Supports

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    Scaffolding, Active Engagement, Modeling, Formative Assessment
    W: The focus of the lesson is probability, different kinds of probability, and the likelihood of events. 
    H: Use the Stations Walk activity to get students thinking about and discussing the likelihood of different events. 
    E: Use the Color Mystery activity to differentiate between experimental probability and theoretical probability. Discuss equally likely and unequally likely events involving the colored blocks from the activity. Introduce the concept of complementary events and discuss equally likely complementary events. 
    R: Use the Carnival Ducky activity or a similar scenario as an opportunity for students to rethink and clarify conceptual ideas such as probability, equally likely, and unequally likely. 
    E: The Partner Quick Quiz can be used to evaluate students’ understanding. 
    T: Tactile activities like randomly choosing blocks from a bag, recording and comparing data, and displaying data in a bar graph get students involved in learning concepts from multiple points of entry. The lesson may be adjusted as necessary using the ideas in the Extension section. 
    O: The lesson is organized so that after a quick review of the difference between experimental and theoretical probability, students participate in a hands-on experiment where everyone has some input into the conclusions drawn. Trends are identified and predictions are made using the students’ own data. As a group, students realize that as the data set gets larger, the experimental probability gets closer to the theoretical probability, which is what the law of large numbers says. 

Instructional Procedures

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    Hand out two or three vocabulary journal pages (M-7-2-1_Vocabulary Journal Page.doc) to students to keep in their folders or binders. Discuss the importance of understanding and using the correct vocabulary words to communicate mathematical ideas clearly. Explain to students that they will be keeping a vocabulary journal throughout this unit. Anytime a new vocabulary word is used, they can add it to their journals. At the end of each lesson, students should review and update the new terms as necessary in their journals.

    “In today’s lesson we will discuss why two or more events are equally likely or why they are not. We will look at probability situations and calculate the probabilities. We will compare two types of probability and how they are related. We will practice strategies that will help us understand more complex probability situations in the next lesson.”

    Prepare ahead of class by writing one pair of events from the Stations Walk transparency (M-7-2-1_Stations Walk.doc) on each sheet of chart paper, and post the sheets around the room. Have the Stations Walk transparency displayed when students arrive.

    Think of a question about yourself that students would be familiar with. You could use favorite things, subjects you teach, marital status, age, etc. For example:

    • If students know a teacher lives a few blocks from school, s/he could ask, “Is it equally likely or unequally likely that I will walk home or drive home from school today?”
    • “Is it equally likely or unequally likely that I like doing math problems or I don’t?”

    Encourage students to share and discuss the reasoning and relevant information they considered when making their decision. Have a brief discussion about the definitions of equally likely and unequally likely events before beginning the Stations Walk Activity. Modify the definitions as necessary.

    Separate students into eight groups to begin a Stations Walk (one group per station).

    “I will be directing you to several stations for this activity. Please bring one marker per group. At the first station, your group will read the events listed, record at the top of the work space whether your group believes the events are equally likely or unequally likely, and provide a reason.

    “Remember to leave room for other groups to write below your comments. At my signal, you will rotate clockwise to a new station and write your response and reason below the previous group’s response.” Have students rotate through about four stations before the class discussion.

    “When you finish with the last station, I will call on someone in your group to read the events and someone to summarize all of the responses. You will have about one minute to work at each station.”

    As groups summarize the responses on their posters, ask questions, and engage the class in discussions to get them thinking critically about each situation and what could have helped them make decisions about which were equally likely. Some of the situations may seem obvious (for instance, problems 2, 5, and 6). However, encourage any students who disagree to argue their point and present specific examples. This type of discussion can be a powerful way to reinforce the understanding of equally likely and unequally likely. To finish, have students write the definition of equally likely on the poster. Students should remain in their groups for the next activity.

    In the Color Mystery activity students will collect data and make predictions about what is in a container. They will also be asked to adjust the contents of the container to make specific outcomes equally likely.

    Prepare one bucket or paper bag (lunch-bag size) for each group. (They should not be transparent containers.) Inside each container place 24 colored blocks, counters, folded color cards (M-7-2-1_Color Cards.doc), or other uniformly-shaped colored objects. Use 12 blue, 8 red, and 4 yellow objects for each set of 24. Make paper copies and transparency copies of Color Mystery Record Sheet (M-7-2-1_Color Mystery Record Sheet.doc) and Color Mystery Graph (M-7-2-1_Color Mystery Graph.doc) for each student. Do not reveal any information regarding colors or amounts to the students—just the type of objects should be known.

    “Can anyone explain the difference between experimental and theoretical probability?” Discuss briefly.

    “The first task within your groups is to collect experimental data. Without looking at what is in your container, draw out one (name the object you placed in the containers) and record the color on the record sheet in the Data Set 1 section. Replace the (object), and continue until you have selected and replaced 20 times. Each time you select, make note of the color on your record sheet. Stop after 20 tries, and try to determine if each of the colors you drew is equally likely.” Do this activity for about 3 to 4 minutes.

    “What colors do you think are in the container?

    “Do you think there is an equal number of each color in the container?

    “Do you think each color is equally likely to be drawn next? How do the relative numbers of each color affect your prediction of the outcome?

    “What do you think the probability of selecting _____ next is? Why?”

    The complete contents of the container should still not be known at this time.

    Ask the groups to share their findings. Ask the class: “Do all of your colors appear to be equally likely? Why or why not?

    “All of your containers have identical contents. Why do you think the results of the groups are so different?” Students should realize that the outcomes of experiments will differ. If they do not, suggest doing more trials.

    Ask, “Based on all of the groups’ results, who wants to make a prediction about the ratio of colors in our containers?” Record a few predictions on the board and go back to them later when the contents are revealed.

    “Would it help to have more data to determine the ratio of colors and whether they are equally likely? Why or why not?

    “Each group will collect more data. Draw 20 more (objects) to complete the second data set.

    “Add the total number of each color to the table on your record sheet.

    “Compute the fractions and percentages for your two sets of combined data. Finally, answer only questions 1 through 3.” Write the instructions on the board or circle the parts on the overhead for students who do not process multiple instructions easily.

    When students are finished with questions 1 through 3, reveal that the total number of blocks in the container is 24. Have students answer and discuss question 4.

    Before students answer question 5, reveal the exact number of each color of block (12 blue, 8 red, 4 yellow). In the discussion following question 5, mention the fact that the percentages they found for their 40 trials are actually the experimental probabilities. Also, emphasize the equivalent forms of probability (fraction, decimal, and percentage). Discuss what the percentages for the actual numbers of colored blocks represent (the theoretical probabilities). Compare the predictions made earlier to the actual ratios revealed.

    The concept of equally likely should be revisited in relation to the fraction of different colored blocks that appear in the container.

    “How do the fractions (or percentages) of colors relate to the likelihood of choosing a specific color?”

    Give students 1 to 2 minutes to calculate the theoretical probabilities listed below. Check that students understand the form P(event), and what is being asked. Be sure to discuss that “P(not yellow)” is the complement of an event and what this means.

    1. P(red)
    2. P(blue)
    3. P(red or yellow)
    4. P(not yellow)
    5. P(not blue or yellow)

    “Now that you know the actual number of each color, it is evident that the three colors are not equally likely to be drawn. Take a minute to think of a way we could add or remove blocks to make red blocks and blue blocks equally likely to be drawn, without changing the overall number of blocks and without the probability of any color being .” (A variety of correct responses is possible. For example: adding and subtracting enough blocks to end with a total of 1 blue, 1 red, 22 yellow or 2 blue, 2 red, and 20 yellow, etc.)

    The second part of this activity compares the experimental and theoretical probabilities and demonstrates how the law of large numbers is applicable in probability problems.

    “We are going to begin the second part of the activity. You will be combining the data that all of our groups collected. You will need your Color Mystery Graph sheet (M-7-2-1_Color Mystery Graph.doc). I am going to ask each group for the results of the experiments, and we will all record the data on the table, totaling it as we go. This is a good opportunity to do some mental arithmetic, so be prepared to be called on for each new total and do not use a calculator.” This should take about 5 minutes.

    “Next we will be calculating the percentage of each color that was drawn. We will begin with the first 40 trials and we can start with blue. We saw that the number of blue items drawn was XX and the total number of trials was 40. So we divide XX by 40 to get the percentage of blue items drawn. Next we will do the same thing for red. We know that there were YY red items drawn in the first 40 trials, so we take YY divided by 40 to find the percentage of red items drawn in the first 40 trials. You will do the same for yellow.

    “The first space in the 80 column shows the total number of blue items drawn, ZZ, in 80 trials. So we will take ZZ and divide it by 80 to find the percentage of blue items drawn in 80 trials. During work time, complete the table.

    “Before we begin this step, I would like you to take 3 minutes to discuss with your group what you think our class results will be. Will they be different than your results? Which specific information or data will affect your prediction?” Ask some groups to share their responses.

    After students have worked for a few minutes, interrupt and demonstrate graphing the first column or two.

    “After you have finished the table, go ahead and graph the percentages from the table.” Give students about 10 to 15 minutes to complete the table and graph. This could also be assigned as homework if time does not permit it to be completed during class time.

    When students are finished, ask them to share their observations. The discussion should lead to the conclusion that the more sets of data that were included, the closer the experimental probability came to the theoretical probability. Explain that this is the basis of the law of large numbers. Use some specific examples such as the difference between the outcomes of 10 flips of a fair coin and 10,000 flips of the same coin.

    This is an excellent opportunity to utilize a computer-generated simulator like the one at http://www.shodor.org/interactivate/activities/Marbles/. The simulator draws marbles from a bag. Adjust the numbers of the different colors to the experiment numbers. Begin with 10 trials, then 50, then 100, then 1000, and so on. Have students pay close attention to the differences in experimental and theoretical probabilities. When you reach a very large number of trials, observe that even with an extremely high number of trials, the experimental probability continues to fluctuate above and below the theoretical probability. This could be an extension activity too, as stated below. Also show that every time a fixed number of trials is performed, the probability is different, just as the class groups had different outcomes.

    Present a new scenario to the students. Use the one below or create your own.

    The school carnival has a game with 120 rubber duckies floating in a pool. Each is marked on the bottom with a phrase indicating what prize, if any, a person wins by selecting that duck on his/her turn. The ducks will be placed back in the pool after each person’s turn, so there will always be 120 ducks in the pool when a player makes his/her selection. The ducks are marked as follows:

    • 40 are marked “no prize.”
    • 20 are marked “stuffed animal.”
    • 12 are marked “small toy.”
    • 8 are marked “$10.”
    • The rest are marked “candy bar.”

    Ask students to record answers to the following questions or similar questions related to a scenario of your choice. Suppose 80 people played the game (selected one duck to win a prize):

    • “Are there any outcomes which are equally likely to be selected?” (No prize and candy bar because each have 40.)
    • “Which prize do you predict will be won the most often?” (Candy bar because it has the most markings other than no prize.)
    • “What is the probability of that prize being selected?” (There are 40 out of the total of 120 ducks, so  or .)
    • “How many times out of 80 do you predict that it will be selected?” (Since the probability is , 80 ÷ 3 is about 26 or 27 times)
    • “Which prize do you think will be selected least often?” ($10 because there are only 8 ducks marked for this)
    • “What is the probability of this prize being selected?” (There are 8 out of 120 ducks in total, so  or .)
    • “How many times out of 80 do you predict it will be selected?” ( of 80 times, so 80 ÷ 15 is between 5 and 6)
    • “Find the probabilities for the remaining prizes.”

     P(No prize) = ; P(stuffed animal) =  or ; P(small toy) =  or

    • Challenge question for any group who finishes early: “How many of each prize should they have ready if they expect 100 people to play the game?”

    Stuffed animal:  = 16.666 or about 17%, so 17 (out of 100 tries)

    Small toy:  = 0.10 or 10%, so 10 (out of every 100 tries)

    $10:  = 0.0666 or about 7%, so 7 (out of every 100 tries)

    Candy:  = 0.333 or about 33%, so 33 (out of every 100 tries)

    No prize for 33 people.

    Check: the total is 100

    • Challenge question for students who are at or going beyond the standard: “How many of each prize do you think they should have ready if they expect 180 people to play the game?”

    Stuffed animal: 30

    Small toy: 18

    $10: 12

    Candy: 60

    No prize for 60 people.

    Check: the total is 180

    Allow approximately 15 to 18 minutes to complete the activity. Circulate around the room asking students guiding questions and clarifying any misconceptions. Let students know that two or three groups will be randomly selected to present their answers.

    Throughout the Stations Walk and Part 1 of the Color Mystery activity, students should be continually given opportunities to discuss, ask questions, and revise their work. Part 2 of the Color Mystery activity offers students a chance to rethink and clarify conceptual ideas such as probability, equally likely, and the law of large numbers.

    To evaluate students’ understanding of concepts from this lesson, have students work for about 5 to 10 minutes on the Partner Quick Quiz (M-7-2-1_Partner Quick Quiz and KEY.doc).

    Extension:

    • Routine: Students share ideas by working on problem-solving strategies with partners and groups. Emphasis is placed on communicating probability concepts using correct vocabulary. Students will create a vocabulary journal. Throughout the lesson, students will add to their journals a definition and an example for each word. In this lesson the following words should be included: outcome, complementary event, experimental probability, theoretical probability, equally likely, unequally likely, and law of large numbers.
    • Small Groups: Students who had difficulty with the Partner Quick Quiz or those still needing opportunities for additional learning can work on a simpler problem to gain greater understanding of the concepts in this lesson.

    An example would be to flip a coin one time. If you get heads, ask students: “Does this mean I will always get heads because 100% of the time (1 out of 1) I got heads?

    “What does it mean if I also flip heads on my second toss? Since it is still 100%, can I be certain it will always be heads?

    “On the third toss I got tails. So does that mean  of the time I can expect heads and  of the time expect tails? What else do I need to consider?” (Students may say something like, “You need to flip it a lot more times before you decide the probabilities.”)

    “I am going to toss a coin ten times. Predict how many heads and tails you think I will get. Write it down on a piece of paper.” Toss the coin and compare the results to the student predictions. Ask why they predicted the way they did. Take the opportunity to explain that over a large number of trials, the theoretical probability of heads and tails is 50% for each.

    Have students flip coins themselves and keep a record. After every five (or ten) trials have them figure out the probabilities for heads and tails. Discuss how their probabilities get closer to 50% each time as they conduct more trials. You could have them use a random-number generator (http://www.random.org/integers/) or coin flipper (http://www.random.org/coins/) on a computer instead of flipping coins.

    • Expansion 1: For students who did well on the Partner Quick Quiz and showed proficiency during the block activity in the lesson, use the following extension. Continue working with the container of blocks from the lesson. “Using the addition of blocks to your container (without removing any blocks), figure out how to make drawing red and blue blocks equally likely, without changing the current probability of selecting yellow. Examine your answers and describe the relationship between the new numbers of blocks of each color and the original numbers of blocks of each color.”

    (Possible solution: Add 3 blue, 7 red, and 2 yellow to get the overall totals 15 blue, 15 red, and 6 yellow (blue  , red  , yellow  ). Other solutions include any number of block additions that create multiples of the overall totals in the first solution.)

    • Expansion 2: To challenge students who displayed proficiency and advanced thinking during the lesson, use the Carnival Ducky Pool scenario from the lesson, but add the following information:

    “The carnival committee expects about 220 students to play the game and each to pay $1.00 to play. The prizes for the game cost the following amounts:

    • Candy = $0.25
    • Small Toy = $0.65
    • Stuffed animal = $1.10
    • $10 bill = $10”

    Have students answer the following questions:

    • “How much money will the Ducky game earn from ticket sales?” (220 x 1.00 = $220)
    • “How many of each prize should the committee expect to give away?”

    No prize: about 73

    Candy: about 73

    Small Toy: 22

    Stuffed animal: about 37

    $10: 15

    Check: total is 220

    • “How much will they spend purchasing these prizes? Show your work.”

    Candy: about 73 x 0.25 = $18.25

    Small Toy: 22 x 0.65 = $14.30

    Stuffed animal: about 37 x 1.10 =$40.70

    $10: 15= 15 x 10 = $150

    • “Will the committee earn enough from the 220 tickets they plan to sell to pay for the prizes you predict they will need to give away?” (No, because the total earned is $220, and the total cost of prizes is $223.25.)
    • Technology Connection: http://www.shodor.org/interactivate/activities/Marbles/

    This site has a colored-marble simulation activity that is a good summary for the law of large numbers. Use the classroom computer to display the simulation or have students work on computers and do their own simulations. For an extension, students can study what happens when the ratios of the colors are changed, more or fewer colors are used, and the number of marbles drawn at one time is changed.

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Final 07/19/2013
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