In this activity, students are invited to use different methods for writing out the sample space for spinning three different spinners. After trying a list, table, and tree, students are asked to compare the methods and describe the benefits and drawbacks of the methods.

Students should use the structure of the organization methods to decide which methods they like to use best (MP7).

Some students may create a tree diagram but may not understand how to quantify the number of outcomes in the sample space. Prompt students to look at their tree diagram and to count each individual outcome. Show students that each branch at the end of the tree represents an outcome in the sample space.

The purpose of this discussion is for students to understand how each of the three representations can be used to represent the same sample space.

Here are some questions for discussion.

• “How many cells will there be in Tyler's method for each table?” ( first, then )
• “How many paths are in Lin’s method?” (The first level of the tree has 2 groups of 1 thing each. The second level has 2 groups of 3 things each. The third level has 6 groups of 4 things each. The number of paths is equal to the number of endpoints of the paths, and there are of these.)
• “Which method do you think you would use to find probabilities of events for this experiment? Explain your reasoning.” (I would use the organized list. I just start with the sample space from the first spinner and then add to the list using the sample space from the second spinner, and so on. It is similar to a tree diagram, but I don’t actually create the tree diagram on paper.)
• “How did you find the probabilities in this activity?” (I divided the number of outcomes in the event by the number of outcomes in the sample space.)

Tell students that probabilities can only be found in this way when each outcome is equally likely. If the section labeled A were larger than the section labeled B, then another method would need to be used to determine the probabilities of events. This will be addressed later in the unit.

If time permits, display each representation for all to see.

Tree diagram:

Here are some questions for discussion.

• “What are the benefits of each representation?” (The organized list summarizes the same information that is in the table and the tree diagram, and it is easy to see all of the information at once. The table organizes the information so that you can see where it comes from. The tree diagram helps you to see that you have all of the possibilities.)
• “Why wouldn’t you use one representation over another?” (I prefer to use the organized list over the tree diagram because it is hard for me to see each of the combinations in the tree diagram. However, sometimes I do use the tree diagram to help me create my organized list.)
• "How could you determine the new number of outcomes without writing out the sample space when we add the fourth spinner?" (I would double the number of outcomes because I know that there are two additional possibilities for each existing outcome.) 

In this activity, students continue using different methods to record the sample space for experiments involving multiple steps. Monitor for students who use these approaches:

• Organized list
• Tables
• Trees
• Other methods for writing the sample space

The purpose of this discussion is for students to analyze different methods for recording sample spaces and to determine probabilities from a sample space using different representations.

Display 2–3 methods for writing the sample space from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different methods. Here are some questions for discussion:

• “What do the methods have in common? How are they different?”
• “How do the different outcomes show up in each method?”

Here are some questions for discussion.

• “For the experiment with the number cube and the coin, what is the probability that the outcome includes heads? Explain your answer using the representation you used.” (The probability is . I made a tree diagram and for every roll of the number cube, half of the outcomes had heads and half of them had tails.)
• “For the experiment with multiple coins, what is the probability that the outcome is three heads and one tail? Explain your answer using the representation you used.” (The probability is . I created an organized list. After writing HHHH, I recorded all the possible outcomes that had exactly three heads. There are 4 of them out of 16 outcomes in the sample space and each outcome is equally likely, so the probability is or equivalent.)
• “For the experiment with the spinners, what is the probability that the outcome is R1, R2, G1 or G2? Explain your answer using the representation you used.” (The probability is or equivalent. I looked in my table, and each of the 4 outcomes listed appeared only one time, so there are 4 outcomes in the event and 20 outcomes in the sample space, and each outcome is equally likely.)