Students should have a basic understanding of experimental probabilities from prior lessons. Theoretical probability will be introduced through a range of activities and compared with experimental probability. Students will: 
- understand the various methods for finding a sample space.
- calculate theoretical probability using the sample space.
- investigate simple probabilities both experimentally and conceptually, and understand the differences in calculations. 
- understand that experimental probabilities are based on experiments, and theoretical probabilities are based on theory. 
- determine the situations in which theoretical probabilities can be used to make decisions. 
- determine whether the outcomes are equally likely or not based on theoretical probability.
 

- What does it mean to analyze and estimate numerical quantities? 
- What makes a tool and/or strategy suitable for a certain task? 
- How may data be arranged and represented to reveal the relationship between quantities? 
- How does the type of data effect the display method? 
- How can probability and data analysis be used to make predictions? 
- How are mathematical properties of objects or processes measured, calculated, and/or interpreted? 
 

- Compound Event: Two or more simple events.
- 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 landing heads-up and the coin landing tails-up—are equally likely to occur.
- Likely Event: The event that is most likely to happen. The probability of a likely event is generally between \(1 \over 2\) and 1.
- Outcome: One of the possible events in a probability situation.
- Probability: A number between 0 and 1 used to quantify likelihood for processes that have uncertain outcomes (such as tossing a coin, selecting a person at random from a group of people, tossing a ball at a target, or testing for a medical condition).
- Proportion: An equation showing that two ratios are equal.
- Random Sample: A sample in which every individual or element in the population has an equal chance of being selected. A random sample is representative of the entire population.
- Sample Space: The set of possible outcomes of an experiment; the domain of values of a random variable.
- Simple Event: One outcome or a collection of outcomes.

- student copies of Vocabulary Journal pages (M-7-1-1_Vocabulary Journal) 
- 10 to 15 pairs of six-sided number cubes 
- one or more pair(s) of four-sided number cubes 
- 50 colored blocks, marbles, or number cubes (2 red, 8 blue, 15 yellow, 25 green) 
- markers for student use 
- chart paper or poster board for each group of four students, two sheets per group 
- student copies of the Partner Quiz (M-7-1-1_Partner Quiz and KEY) 
- student copies of the Marbles Rock activity sheet (M-7-1-1_Marbles Rock and KEY) 
- a variety of marbles, number cubes, and cards for students to use designing a game 
- student copies of the Short Stack activity sheet (M-7-1-1_Short Stack and KEY) 
- copies of Group Spinner sheet (M-7-1-1_Group Spinner Presentation and KEY) 
- student copies of the blank Color Chart (M-7-1-1_Color Chart)

- Observing the Think-Pair-Share activity while calculating a complimentary event should assist identify the student's level of knowledge. 
- Student development will be assessed based on their performance in the Group Spinner activity. 
- The Partner Quiz (M-7-1-1_Partner Quiz and KEY) will assess students' comprehension of theoretical and experimental probabilities.
 

Scaffolding, Active Engagement, Metacognition, Modeling, Formative Assessment 
W: Teach students about experimental probability, calculating experimental probabilities and determining if outcomes are equally likely or not. 
H: Have students predict and track results of spinner events, expressing the results as ratios and in terms of equally likely or not. 
E: Divide students into groups to participate in a number cube rolling activity and display their results in class. 
R: Clarify any misconceptions students may have from the number cube activity. Have all groups combine their data and calculate experimental probabilities using the new totals. 
E: Evaluate students individually and suggest exercises based on their comprehension level. 
T: Encourage students to identify instances of chance and prediction throughout the school year. Some students may need additional tasks to understand the terms 'numerator' and 'denominator' and how they apply to probabilities. Additional probability tests can be used to reinforce probability theory. 
O: This lesson introduces students to experimental probabilities, demonstrates how to calculate small and large trials, and discusses how these factors affect the usefulness of probabilities. 

The purpose of this lesson is to introduce students about the fundamental differences between experimental and theoretical probabilities. The spinner scenario allows students to compare the two categories of probability. Students learn to identify all possible outcomes in a range of situations by creating multiple sample space models. Students should be introduced to the idea that experiments based on a large number of trials would produce more consistent results, similar to the outcomes using theoretical probability. This will get students ready for the formal introduction to the law of large numbers in Lesson 3. Containers containing colored blocks or objects provide a context for a wide range of experimental and theoretical investigations and questions. Questions should include determining theoretical probabilities, calculating the sum of the probabilities of all possible outcomes, recognizing the relationship between a probability of 1 and a specific event, and asking about complementary events (events that do not occur). Explain how the probabilities of all events added together equal 1. For example: If you have a number cube numbered from 1 to 6, the probability of rolling each number is \(1 \over 6\). The sum of the probability of rolling any of the numbers is \(1 \over 6\) + \(1 \over 6\) + \(1 \over 6\) + \(1 \over 6\) + \(1 \over 6\) + \(1 \over 6\) = 1; it is certain to occur.

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

Without telling students exactly how many of each color are in the jar, start drawing samples. After drawing each block, instruct students to color one square above the relevant color in the graph. Return the block, shake it up, and draw again. Continue drawing until you have approximately 20 or 25 blocks. Students' graphs should look similar to 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?" Allow students to share their ideas. (They should point out that the ratio of colors from the trials displays 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 additional block. If it's green, explain that it's what they expected because based on experience (experimental data), green has been drawn the most often so far. If green is not drawn, remind students that our choice is still random, but over time we may discover a predictable pattern (i.e., 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."

Distribute two vocabulary journal pages to students to keep in their folder or binder. Explain that they will keep a vocabulary journal throughout the unit. When students learn a new vocabulary word, they can record it in their journal. Discuss how important it is to comprehend and use the appropriate vocabulary words while communicating mathematical ideas. At this point, ask them to add the terms equally likely, experimental probability, and outcome. Have students discuss their responses with a partner before checking them as a group. At the end of each class, review the new terms in student journals and update them as needed.

"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 calculating the experimental probability by dividing the total number of blocks drawn by the specific colors used.) 

"Let's calculate the experimental probabilities for each color. How many times did I draw out a block?" (20 in this example) 

Allow students to offer aid in completing the probabilities in several forms. Review using ratios to calculate probabilities in both decimal and percentage form. [Note: responses are based on the sample provided but may most likely differ in your own classroom.]

P(red) = \(1 \over 20\) = 0.05 or 5%

P(blue) = \(4 \over 20\) = \(1 \over 5\) = 0.2 = 20%

P(yellow) = \(5 \over 20\) = \(1 \over 4\) = 0.25 = 25%

P(green) = \(10 \over 20\) = \(1 \over 2\) = 0.5 = 50%

Next, reveal the actual contents of the jar to the students a few blocks at a time. Allow them to fill in the bars in the graph next to the experimental bars with the exact number found 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 in the jar. What conclusions can you draw about how they compare?" Students may make observations like these:

Each bar is higher than in our previous testing. 
The bars are all about twice as high as 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 more blue than 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) = \(2 \over 50\) = \(1 \over 25\) = 0.04 or 4%

P(blue) = \(8 \over 50\) = \(4 \over 25\) = 0.16 = 16%

P(yellow) = \(15 \over 50\) = \(3 \over 10\) = 0.3 = 30%

P(green) = \(25 \over 50\) = \(1 \over 2\) = 0.5 = 50%

"These are called the theoretical probabilities. They are based on known quantities or facts rather than being based on experimental data. They are still probabilities; in other words, they indicate 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?" (similar, but not exactly the same) 

"We'll look at another case using coins. What are the possible results of tossing one coin?" (heads or tails) 

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

Students are likely to mention only two or three of the outcomes listed above because they do not always consider heads on the first coin with tails on the second coin as a significantly different outcome from tails on the first coin with heads on the second coin. If required, clear up this misperception. 

"This is our sample space. It is a list of all possible outcomes while tossing two coins. We can organize our data in numerous ways to ensure that we have considered all possible outcomes. The first is an organized list."

"Start with one event, such as 'heads.' List the heads for each event that could be matched to it. Then list the next possible first event, in this case 'tails,' and repeat for each possible second event. Our list would be as follows: heads/heads, tails/heads, tails/tails."

"We might also use parenthesis and/or list the outcomes vertically, as 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 may have taken to determine our sample space in the coin problem. It's an organized table, like a multiplication table." Note that the sample space includes the part of the table that is highlighted gray. These are the same four outcomes mentioned in the organized list.

"The third method is a counting tree (or tree diagram), which yields the same sample space. When calculating theoretical probability, keep in mind that the sample space (list of all possibilities in the diagram's right column) is used to count the number of favorable outcomes as well as the total number of outcomes when calculating theoretical probability." Show a tree diagram like the one below:

"Once you've identified all possible outcomes for a situation, you can use them to calculate various theoretical probabilities. The probability of receiving a specific outcome is calculated using the following ratio:

When expressed as a fraction or decimal, each probability has a value ranging from 0 to 1. When the same values are expressed in percentage form, the values range from 0% to 100%. For example, ¼ = 0.25, or 25%." 

Consider the following examples: 

P(2 heads) = \(1 \over 4\) or 25% (since there are four outcomes and only one is favorable) 

P(a head and a tail) = \(2 \over 4\) = \(1 \over 2\) = 50% (two out of four outcomes have one head and one tail) 

P(no tails) = \(1 \over 4\) = 25% (only heads/heads works because the other three outcomes contain tails at least once; discuss complementary event)

P(no heads and no tails) = \(0 \over 4\) or 0% (discuss impossible event) 

P(2 heads, 2 tails, 1 of each) = \(4 \over 4\) = 1 or 100% (discuss specific event)

Use this opportunity to discuss the impossible event, a specific event, and the complement of an event. Explain that impossible events have a 0% chance of occurring, but some events have a 100% chance of occurring, and that the complement means the likelihood of the event not occurring. Another way to think about complement is the possibility of any other outcome occurring except this specific one.

Think-Pair-Share: Give them one minute to come up with another complement question for the coin problem and the correct answer. Have them turn to a partner and attempt to answer each other's question. Randomly select a few pairs to share their examples with the class. Clarify any misconceptions. 

After the Think-Pair-Share, assign each pair of students to 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 divide each method among half of the groups or let them choose). "Each group will present their table or tree on a piece of chart paper. On a second sheet of chart paper, answer ten probability questions based on your group's sample space. For example, when you mix (red, red), you get red, whereas (red, blue) yields purple. A match would be an outcome like (yellow, yellow). Each group will be given around 20 to 25 minutes to present their findings."

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

Distribute copies of the Group Spinner sheet (M-7-1-1_Group Spinner Presentation and KEY), and tell groups to collaborate to complete the worksheet. 

While student groups are working, walk around the room and observe group discussions and work done on their posters. Ask leading questions to help students avoid misconceptions and incorrect answers. Encourage students to adjust their work as needed. Allow students to make modifications if their student presentations indicate a need.

Pair students together after they have completed the lesson's activities. Each pair will take the Partner Quiz together to provide extra feedback on their grasp and mastery of the theoretical probability concepts covered in this lesson (M-7-1-1_Partner Quiz and KEY). The quiz should take about 10 to 12 minutes to complete. 

Extension: 

Use these suggestions to personalize this lesson to your students' needs throughout the unit and year.

Routine: Discuss the significance of understanding and using the appropriate vocabulary words to communicate mathematical ideas clearly. During this lesson, students should record the following terms in their 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 that 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 to they can relate to. Ask them to calculate the probability of selecting one item from the possible combinations. For example: 

Find all of the possible combination for the entrée, side dish, and dessert offered available in the cafeteria today. 
What are all the possible combinations of outfits you might have worn to school today if you had ____ pairs of pants, ____ shirts, and ____ pairs of socks? What is the probability of choosing a pair of pants from a pile of clothes on the floor?
Find all of the basketball uniform options for our school team if the shorts are _____ or _____, the shirts are _____ or _____, and the socks are _____, _____, or _____. 

Small Group: Short Stack Activity
This activity is for students who need help identifying a sample space and/or calculating theoretical probabilities. Divide the students into groups of two or three.

This project requires two small piles of playing cards for demonstration reasons only. You can use cards from a standard 52-card deck or create a paper set. One pile of three cards is labeled (2, 3, 4), and another pile of three cards is labeled (3, 5, 7). Show how to mix each set of three cards (keeping them in two separate piles), then draw one card from each pile, record it, and return 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 type of pair, experimental probability of receiving specific 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 al 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). 

"You and your partner(s) will complete a tree diagram, often known as a counting tree, as well as an organized table. Each will help you identify all of the possible outcomes for this situation. This list of all possible outcomes is known as the sample space. Use your sample space to determine the theoretical probabilities stated at the bottom of the activity sheet. Remember that probability is calculated 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 selecting different values from the sets of cards, or have students consider a pair of four-sided number cubes for further practice. 

Expansion 1: Make a tree diagram (or counting tree) to determine all possible outcomes while choosing a three-item meal at the roller rink concession stand. Students may choose: 
Entrée: corndog or hamburger. 
Side dish: fruit, chips, or French fries. 
Drink: water, juice, or soda. 
Additional questions could include:

"Explain how the list of possible outcomes can help you calculate the theoretical probabilities of certain meals being selected. Provide 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 only shows two, 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 basically describe the Fundamental Counting Principle. The Fundamental Counting Principle states that if an event may 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 can also 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 option to the drinks? Explain."

Partner Extension: Got Game? Activity 
Create a basic game with marbles, number cubes, spinners, or cards, either with a partner or in small groups. Include rules and a scoring system based on the possibility of each outcome occurring, ensuring that each player has a chance to win. Play the game to demonstrate the experimental probability of the outcomes occurring. Also, explain how you calculated the theoretical probability of each outcome and how it relates to the scoring system. Introduce 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 visit http://www.shodor.org/interactivate/activities/Marbles/ and complete the Marbles Rock activity sheet (M-7-1-1_Marbles Rock and KEY). This can also be done as a class activity if you can project the link from your computer.