• Observation during the Think-Pair-Share activity while calculating a complementary event should help to determine student level of understanding.
• Performance on the Group Spinner activity will serve as a basis for assessing student progress.
• Partner Quiz (M-7-1-1_Partner Quiz and KEY.docx) will measure student understanding of theoretical and experimental probabilities.

The goal of this lesson is to introduce students to the fundamental differences between experimental and theoretical probabilities. Students compare the two types of probability using the spinner scenario. Students learn to identify all possible outcomes for a variety of situations through the creation of multiple sample space models. Students should be introduced to the idea that experiments based on a large number of trials will have more consistent results similar to the outcomes using theoretical probability. This will prepare students for the formal introduction to the law of large numbers in Lesson 3. Containers with colored blocks or objects serve as a context that allows for a variety of experimental and theoretical investigations and questions. Questions posed should include finding theoretical probabilities, finding the sum of the probabilities of all possible outcomes and recognizing the relationship between a probability of 1 and a certain event, and asking questions about complementary events (events not occurring). Explain that the probabilities of all the events added together equals 1. For example: If you have a number cube numbered from 1 to 6, the probability of rolling each particular number is . The sum of the probability of rolling any of the numbers is  ; it is certain to happen.

Prepare a nontransparent jar by placing 50 colored blocks (marbles or number cubes will work too) in it. One possible combination of colored blocks might include 2 red, 8 blue, 15 yellow, and 25 green blocks. As students enter the classroom, hand them the Color Chart (M-7-1-1_Color Chart.docx).

Without telling students exactly how many of each color are in the jar, begin drawing out samples. After each block is drawn, have students color in one square above the corresponding color in the graph. Return the block, shake them up, and draw again. Continue this until you have drawn about 20 or 25 blocks. Students’ graphs should look something like the one below, based on the blocks that were drawn.

“Based on the blocks that were drawn, what conclusions can you make about the contents of the jar?” Let students share their thoughts. (They should bring up the fact that the ratio of colors from the trials shows there are significantly fewer red and many more green than any other color.)

“Can you tell anything about how many blocks are in the jar altogether?” (no)

“If I draw out one more block, what color do you predict it will be?” (green)

Draw one more block. If it is green, explain that this is what they expected because based on experience (experimental data), green has been drawn most often so far. If green is not drawn, remind students that our choice is still random, but over time we may determine a predictable pattern (i.e., likely more green based on what we know so far).

“Today we will review what we know about experimental probability and we will look at another way to calculate and describe the likelihood of a specific outcome occurring.”

Hand out two vocabulary journal pages to students to keep in their folder or binder. Explain that they will be keeping a vocabulary journal throughout this unit. Anytime a new vocabulary word is used, they can add it to their journal. Discuss the importance of understanding and using the correct vocabulary words to communicate mathematical ideas clearly. At this time, have them add the terms equally likely, experimental probability, and outcome. Have students discuss their responses with a partner and check them as a group. At the end of each lesson the new terms should be reviewed and updated as necessary in student journals.

“If you look at the colored block bar graph, how can you find the likelihood or probability of drawing a certain color?” (Students should suggest finding the experimental probability by making ratios using the specific colors drawn to the total blocks drawn.)

“Let’s calculate our experimental probabilities for each color. How many times did I draw out a block?” (20 in this example)

Have students offer assistance in completing the probabilities in multiple forms. Review using the ratios to find the probabilities in decimal form as well as percentage form. [Note: answers are based on the sample provided but will likely vary in your own classroom.]

• P(red)=  = 0.05 or 5%
• P(blue)=  = 0.2 = 20%
• P(yellow)=  = 0.25 = 25%
• P(green)=    = 0.5 = 50%

Next, reveal the actual contents of the jar to the class a few blocks at a time. Have them fill in the bars in the graph next to the experimental bars with the exact number contained in the jar (2 red, 8 blue, 15 yellow, and 25 green).

“Compare the bars on your bar graph from our experiment to the actual number of cubes that were in the jar. What observations can you make about how they compare?” Students may make observations similar to these:

• Each bar is higher than for our trials.
• Bars are all about twice as high for the actual blocks.
• There are still fewer red and more green than the other colors.
• There are about twice as many yellow as blue.
• There are four times as many blue as red.

“Now that we know exactly how many cubes of each color were in the jar, let’s calculate the probabilities again using these numbers.”

• P(red)=  = 0.04 or 4%
• P(blue)=   = 0.16 = 16%
• P(yellow)= = 0.3 = 30%
• P(green)=    = 0.5 = 50%

“These are called the theoretical probabilities. Instead of being based on experimental data, they are based on known quantities or facts. They are still probabilities; in other words, they represent what we expect to happen, but chance is still in play when we draw.”

“How do the experimental and theoretical probabilities for each color compare?” (close, but not quite the same)

“We are going to look at another example involving coins. What are the possible outcomes if I toss one coin?” (heads or tails)

“What are the possible outcomes if I toss two coins?” (head-tails, tails-heads, tails-tails, or heads-heads)

It is likely that students will only bring up two or three of the outcomes above because they do not always consider heads on the first coin with tails on the second coin as a distinctly different outcome from tails on the first coin with heads on the second coin. Clarify this misconception if necessary.

“This is called our sample space. It is the list of all possible outcomes for tossing two coins. There are several ways we can organize our data to be sure we have accounted for all possible outcomes. The first is an organized list.”

“Start with one event, like ‘heads.’ List heads with each event that could be matched with it. Then list the next event that could be first, in this case ‘tails,’ and list it once for each event that could happen second. Our list would look like this: heads/heads, heads/tails, tails/heads, tails/tails.”

“We could also use parentheses and/or list the outcomes vertically so it would look like this:

(heads, heads) (heads, tails) (tails, heads) (tails, tails)

or

heads/heads

heads/tails

tails/heads

tails/tails

In each case you can see four distinct outcomes.”

“Let’s look at another method we could have used to find our sample space in the coin problem. It is an organized table, similar to a multiplication table.” Point out that the sample space includes the part of the table that is highlighted gray. These are the same four outcomes as identified in the organized list.

“The third method we can use is a counting tree (or tree diagram), which will also produce the same sample space. Remember the sample space (list of all outcomes in the right column of the diagram) is used to count the number of favorable outcomes and total number of outcomes when calculating theoretical probability.” Demonstrate a tree diagram like the one below:

“Once you have identified all of the possible outcomes for a situation, you can use them to determine different theoretical probabilities. The probability of getting a specific outcome is found using the following ratio:

Each probability will be a value between 0 and 1 when written as a fraction or decimal. The values will be between 0% and 100% when we write the same values in percentage form. For example  = 0.25, or 25%.”

Work through examples such as:

• P(2 heads) =  or 25% (since there are four outcomes and only one is favorable)
• P(a head and a tail) =   = 50% (two out of the four outcomes have one head and one tail)
• P(no tails) = = 25% (only heads/heads works since the other three outcomes contain tails at least once; discuss complementary event)
• P(no heads and no tails) =  or 0% (discuss impossible event)
• P(2 heads, 2 tails, 1 of each) =  = 1 or 100% (discuss certain event)

Take this opportunity to discuss the impossible event, certain event, and the complement of an event. Explain that impossible events have 0% chance of occurring, certain events have 100% chance of occurring, and that complement means the likelihood of the event not happening. Another way to think of complement is the likelihood of any other outcome happening except this specific one.

Think-Pair-Share: Give students one minute to think of another complement question for the coin problem and the correct answer. Have them turn to a partner and try to answer each other’s question. Call randomly on a few pairs to share their examples with the class. Clarify any misconceptions.

After the Think-Pair-Share, match each pair of students with another pair to form groups of four for the next activity.

Group Spinner Activity

“You are going to find the sample space for a double spin on this spinner by using an organized table or counting tree.” (You can assign each method to half of the groups or let them select). “Each group will display its table or tree on a piece of chart paper. On a second sheet of chart paper you will answer ten probability questions using your group’s sample space. For example, when you mix (red, red) you get red, while (red, blue) would mix to purple. A match would be an outcome like (yellow, yellow). In about 20 to 25 minutes each group will be asked to present its findings.”

Provide each group with a variety of markers, ruler/straightedge (optional), and two pieces of chart paper or poster board.

Hand out copies of the Group Spinner sheet (M-7-1-1_Group Spinner Presentation and KEY.docx) and instruct groups to work together to complete the worksheet.

While student groups are working, walk around the room observing group discussions and work done on their posters. Ask leading questions to guide students away from misconceptions and incorrect answers. Encourage students to adjust their work if necessary. Also, allow students to make corrections if their student presentations reveal a need.

After the activities in the lesson have been completed, pair students together. Each pair of will complete the Partner Quiz together to provide additional feedback on understanding and mastery of the theoretical probability concepts presented in this lesson (M-7-1-1_Partner Quiz and KEY.docx). The quiz should take approximately 10 to 12 minutes to complete.

Extension:

Use these suggestions to tailor this lesson to meet the needs of your students during the unit and throughout the year.

• Routine: Discuss the importance of understanding and using the correct vocabulary words to communicate mathematical ideas clearly. During this lesson the following terms should be entered in students’ vocabulary journals: certain event, complementary event, impossible event, model, organized list, organized table, theoretical probability, sample space, and tree diagram. Keep a supply of vocabulary journal pages on hand so students can add pages as needed.

As a class warm-up on several different days, ask students to explain the sample space for a variety of situations they can relate to. Have them determine the probability of selecting one item from the possible combinations. For example:

• Find all the possible combinations for the entrée, side dish, and dessert offered in the cafeteria today.
• What are all the possible combinations of outfits you could have chosen to wear to school today if you have ___ pairs of pants, ___ shirts, and ___ pairs of socks? What is the probability of selecting a pair of pants from a pile of clothes on the floor?
• Find all the combinations of basketball uniforms for our school team if the shorts are _____ or _____, shirts are _____ or _____, and socks are _____, _____, or _____.

• Small Group: Short Stack Activity

This activity is for students requiring further support on finding a sample space and/or calculating theoretical probabilities. Separate students into groups of two or three.

Two small piles of playing cards are needed in this activity for demonstration purposes only. You can take the cards from a standard 52-card deck or make a paper set. One pile of three cards is labeled (2, 3, 4) and another pile of three cards is labeled (3, 5, 7). Demonstrate mixing up each set of three cards (keeping them in two separate piles), drawing one card from each pile, recording it, and returning it to the pile. Ask “If I continued to draw and record the pairs of cards I drew 30 or 40 times, what could you tell me based on the data?” (How many times you got each kind of pair, experimental probability of getting certain pairs, which you got more or less of, etc.)

“Without actually drawing the cards, how can I figure out some of the same things?” (Look at all the ways the cards could be drawn together, make a tree, make a table.)

“Without actually conducting an experiment or doing an activity, I can analyze the set of all possible outcomes to determine the theoretical probability of each outcome happening.”

“It is very important that I use an organized method so that I do not miss any possibilities.”

Give each group the Short Stack activity sheet (M-7-1-1_Short Stack and KEY.docx).

“With your partner(s), you will complete both a tree diagram, sometimes called a counting tree, and an organized table. Each will guide you to find all of the possible outcomes for this situation. This list of all possible outcomes is called the sample space. Use your sample space to find the theoretical probabilities listed at the bottom of the activity sheet. Remember, probability is found by comparing the desirable outcome to the total number of outcomes in your sample space.”

Monitor progress and correct any misconceptions. Create a new example by choosing different values on the sets of cards, or have students consider a pair of four-sided number cubes, if further practice is needed.

• Expansion 1: Make a tree diagram (or counting tree) to find all the possible outcomes for selecting a three-item meal at the roller rink concession stand. Students may select:

• Entrée: corn dog or hamburger
• Side dish: French fries, chips, or fruit
• Beverage: water, juice, or soda

Additional questions may include:

• “Explain how the list of possible outcomes can help you determine theoretical probabilities of certain meals being selected. Include examples.”
• “Explain why an organized table is not a good choice to find the sample space for this problem.” (Because there are three “events” or choices, and the table represents just two with the vertical and horizontal.)
• “What is the theoretical probability of a student selecting a meal containing both a hamburger and a soda?”
• “Describe how you could find the number of possible outcomes without actually drawing the tree diagram.” [Note: Here, students should essentially describe the Fundamental Counting Principle. The Fundamental Counting Principle states that if an event can occur in exactly m ways, and if following it, a second event can occur in exactly n ways, then the two events in succession can occur in exactly (m × n) ways, Similarly, if there are three events that can occur in m, n, and p ways, then these events together can occur in (m × n × p) ways.]
• “How would the number of outcomes change if we added a burrito choice to the entrées, and added a slushy choice to the beverages? Explain.”

• Partner Extension: Got Game? Activity

With a partner or small group, design a simple game using marbles, number cubes, spinners, or cards. Include rules and a scoring system based on the likelihood of each outcome occurring, so that each player has a chance to win. Play the game to show the experimental probability of the outcomes occurring. Also show how you found the theoretical probability of each outcome and how it relates to the scoring system. Present the game to the class or use it as a station activity.

• Technology Connection: Marbles Rock Activity

If student computers are available, have pairs of students go to http://www.shodor.org/interactivate/activities/Marbles/ to complete the Marbles Rock activity sheet (M-7-1-1_Marbles Rock and KEY.docx). This can also be completed as a class activity if the link can be projected from your computer.