In this activity, students learn three different methods for writing the sample spaces of multi-step experiments and explore their use in a few different situations. Since the calculated probability of an event depends on the number of outcomes in the sample space, it is important to be able to find this value in an efficient way. In the discussion, students will explore how different methods may be useful in different situations (MP1).

Monitor for students who use these different approaches for finding the sample space on the last question. Here are some ways students may arrange their thinking from less to more flexible:

• always use a list format to write out the sample space
• always use a tree diagram to write out the sample space
• change which representations they use for different questions

Some students may have trouble interpreting the tree diagram. Help students see that a single outcome is represented by following a branch from the point furthest to the left until they reach the end of a branch on the right side. It may help for students to write the full outcome on the diagram as well. In Priya’s picture, next to the uppermost 1, a student could write H1.

​The purpose of this discussion is to think about the different methods of writing the sample space and when each might be useful. The discussion is also meant to make the connection between the methods and the number of outcomes in the sample space so that students can quickly find the number of outcomes without writing out all the possibilities.

Invite previously selected students to share their strategies for finding the sample spaces. Sequence the discussion of the approaches in the order listed in the Activity Narrative. If possible, record and display their work for all to see.

For students who used the same representation throughout, ask, “Why did you choose to use this same strategy for all the questions? What are the benefits of this strategy? Did you encounter any problems using the strategy?” For students who changed strategies for different questions, ask, “How did you decide which strategy to use for each question?”

Connect the different responses to the learning goals by asking questions such as:

• “How would you represent the outcome R2 for the spinner using each method?” (In a list it would be one of the options written. In a table it would be on the R row and the 2 column. In a tree it is represented by following the branch to R and then to 2.)
• “How do you find the number of outcomes in the sample space using each method?” (Using the list, count each item on the list. From the table, multiply the number of rows by the number of columns to get the number of outcomes in the middle of the table. From the tree, count the number of branch endings.)
• “Are all three of the methods useful for each of the situations?” (The table does not work when there are 3 parts of the experiment, like flipping 3 coins.)

In this activity, students practice using their understanding of ways to calculate the number of outcomes in the sample space without writing out the entire sample space (MP7). Many situations with multiple steps have very large sample spaces for which it is not helpful to write out the entire sample space, but it is still useful to know the number of outcomes in the sample space. In this activity, students find the number of different necklaces that can be made from available options.

Some students may attempt to write out the entire sample space. Encourage them to write a few outcomes to get the idea of what is possible, but let them know that the answer for question 1 is over 100, so finding a pattern or way to calculate the answer might be more efficient.

Some students may not count the none option for pendant as a distinct choice. Show these students that a necklace like “red, large stone, small, single strand” is different than “red, no pendant, small, single strand” and should be counted separately.

The purpose of the discussion is to help students understand the calculations behind the solutions of these problems.

Some questions for discussion:

• “Describe how the tree of necklace options would look without drawing it out.” (The first column could have the 5 options for color. Coming out from each of those options could be 3 branches for each of the pendant options. From each of these there would be 4 more branches for each of the length options. Each of those would have 3 branches for the style.)
• “How is the tree connected to the calculation of the size of the sample space?” (For the choice of color and pendant, there are 5 groups of 3 branches, making 15 options for those two choices. When including the length, there are 15 groups of 4 branches, making 60 options. For the style, there are 60 groups of 3 branches, making 180 options.)
• “If you are allowed a second choice of color for a design on the beads, are there more or fewer options?” (There are more options because each of the 180 options would allow for another choice of 4 options.)