As preparation for talking about probability, students are asked to engage their intuition about the concept by loosely grouping scenarios into categories based on their likelihood. Some of the categories are meant to be loosely interpreted, while others such as “certain” and “impossible” have more precise meanings. Students should be able to construct arguments for their classification (MP3).

The purpose of this discussion is for students to see that the loose categories can be understood in a more formal way. Some students may begin to attach numbers to the likelihood of the events and that is a good way to begin the transition to thinking about probability.

There may be some discussion about the category “equally likely as not.” Are events in this category required to be at exactly 50%, or would an event with a 55% or 48% likelihood be placed in this category as well? At this stage, the categories are meant to be loose, so it is not necessary that everyone agrees on what goes in each category.

Discuss questions such as:

• “What factors make an event difficult to categorize?” (When an event is close to the edge between the categories, it can be hard to label it well.)
• “Which categories are the most strict about what can go in them?” (“Certain” should represent scenarios that must happen with 100% certainty. “Impossible” should represent scenarios that cannot happen.)
• “What does it mean for an event to be certain?” (that it must happen)
• “What does it mean for an event to be likely?” (between about 50% and 100% chance)

In this lesson, students begin to move towards a more quantitative understanding of likelihood by observing a game that has two rounds with different requirements for winning in each round. The game is also played multiple times to help students understand that the actual number of times an outcome occurs may differ from expectations based on likelihood at first, but should narrow towards the expectation in the long-run. By repeating the process many times, students recognize a structure beginning to form with the results (MP8). Activities in later lessons will more formally show this structure forming from the repeated processes.

The purpose of this discussion is to begin moving students towards quantifying the likelihood of events. Invite students to share their answers and reasoning to the last question.

Some questions for discussion:

• “What was the chance experiment in this game?” (rolling the number cube)
• “What number would you assign to the likelihood of evens winning in the first round? What about the likelihood of 1–4 winning in the second round?” (For the first round of the game, there is a 50% or chance for evens to win. For the second round of the game, 1–4 wins about 67% or or of the time.)
• “What percentage or fraction would you assign to waiting for less than 10 minutes before your order is taken at a fast-food restaurant?” (Values between 80% and 100% are expected. It is not necessary to agree on a particular value at this point.)
• “How could we get more evidence to support these answers?” (Collect data from fast-food restaurants to find the fraction of customers who order their food within 10 minutes.)

Students sort different descriptions of chance experiments and events during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

Students may have trouble understanding the “rock paper scissors” context. Tell these students that a player randomly chooses one of the three items to play in each round. If students still struggle, tell them that each of the three items are expected to be played with equal likelihood.

The purpose of the discussion is for students to talk about the methods they used to sort the cards and compare likelihood of different situations.

Some questions for discussion:

• “How are the numerical values of the likelihoods written?” (Some are written as percentages, some as fractions, and some as decimal values.)
• “How do you compare the likelihood of the situations when there is a mix of percentages, fractions, and decimals?” (We convert the percentages to fractions of the form and fractions to decimals by dividing, then order the decimals.)
• “Some of the cards do not have a percentage, fraction, or decimal. How can you determine where those cards would go in the order?” (We think of them as fractions with the number of ways the event might happen in the numerator and the number of ways the chance experiment might result as the denominator.)