In this activity, students describe events as a subset of the sample space, using characteristics of the outcomes. Then they determine probabilities for combined events based on the number of outcomes in the event. Students must reason abstractly and quantitatively to make connections between the countries in the real world and their representation in the diagram (MP2).

Some students may struggle with the difference between "and" and "or" when used in a situation. Ask, “How many students in the school do you think are tired and happy?” Then ask, “If we think about how many are tired or happy, are more students included? What is the difference?”

The goal of this discussion is to help students make the connections between subsets of sample spaces and probability, and to give them additional practice using “and,” “or,” and “not” in mathematics.

Here are some questions for discussion.

• “True or false: All the countries on the diagram are crown dependencies or in the British Isles” (True)
• “How many countries are part of the United Kingdom or are crown dependencies?” (7)
• “If a country is chosen at random, what is the probability that it is part of the Isle of Ireland?” ()
• “What is a country that would go outside of all of the circles in the diagram?” (United States of America, France, China, Ghana, Ecuador)

This activity gives students an opportunity to determine and request the information needed to understand subsets of a sample space. The subsets are then used to find a probability of an event. To complete the tables, students must ask for information from a partner using mathematical language.

The Information Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and to ask increasingly more precise questions until they get the information they need (MP6).

After students have completed their work, share the correct answers,and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “What was challenging about solving the problem?” (The challenging part was having to figure out how to use the information from my partner to fill in the table because the information only gave me clues to find the numbers rather than the numbers themselves.)

• "What percentage of students in your table are in 11th grade?" (37.2% for problem card 1 and 47.5% for problem card 2.)

Highlight for students how a table can be used to represent a sample space and how to calculate probabilities and percentages.

This optional activity provides additional practice understanding combined events. Students roll number cubes to collect data and then analyze the outcomes using language describing characteristics of some outcomes. Then students estimate probabilities based on their collected data.

In the digital version of the activity, students use an applet to simulate rolling number cubes before answering questions about combined events. The applet allows students to collect data quickly, using a number generator. Use the digital version if physical number cubes are not available or if time is limited and students can readily access the applet on a device.

The goal of this discussion is to make sure students understand that probabilities calculated using sample space and probabilities calculated by experimentation are related, but not necessarily equivalent. A secondary goal of the discussion is to informally assess student understanding of how “and,” “or,” and “not” are used when finding probabilities.

Here are some questions for discussion.

• “Describe how you could find the number of outcomes in the sample space for rolling 2 number cubes without writing out all of the possibilities?” (If I picture the tree diagram, there would be 6 numbers on the first level and each would have 6 branches to the second level. The 6 groups of 6 means there are 36 paths in the tree ( ).)
• “According to the sample space there are 36 different outcomes for rolling two number cubes. You rolled the cubes only 20 times. Raise your hand if you rolled the same outcome more than once when you rolled 20 times. Why is this possible?” (It is possible to repeat the same roll more than once because rolling the number cubes is random. Just because you rolled double sixes on one turn does not reduce your chance of rolling double sixes on the next turn.)
• “How did your percentages for the events compare to the actual probabilities? Were any of them close? Why are they not exactly the same?” (Our percentages were close for a few of the listed events, but very different on others. They are not exactly the same because we rolled only 20 times.)
• “How would you expect your percentages to compare to the actual probabilities if you continued rolling all day?” (With more rolls, the percentages should get closer to the actual probabilities.)

Some students may not know how to represent probability as a percentage. Prompt students to multiply the value they obtained to represent probability (between 0 and 1 inclusive) by 100 to get the percentage. Emphasize that a percentage is a quantity described by a rate per 100.