Lesson 2

The purpose of this warm-up is to engage students' intuition about likelihood of events. The following activities in this lesson continue to develop more formal ways of thinking about likelihood leading to the definition of probability in the next lesson.

Arrange students in groups of 2. Give students 2 minutes of quiet work time and time to share their response with a partner. Follow with a whole-class discussion. 

Students may think that it is required to pull out a specific shoe rather than any left shoe. Ask students to visualize the problem and determine how many left shoes are in the closet. 

The purpose of the discussion is to help students recognize their own intuition about the likelihood of an event even when prior outcomes are not available.

Have partners share responses with the class, and ask at least one student for each option for their reasoning. Give students time to discuss their reasoning until they come to an agreement. 

It may be helpful to reiterate that the outcomes from these actions are not certain. It is certainly possible to do both things and draw out a right shoe and listen to the first song on the list first, but it is not very likely based on the situations.

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 by reasoning abstractly about situations in context as in MP2. Some of the categories are meant to be loosely interpreted while others (such as "certain" and "impossible") have more precise meanings.

Arrange students in groups of 2.

Tell students that a “standard number cube” is an object that has the numbers 1 through 6 printed on a cube so that each face shows a different number. This item will be referenced throughout the unit.

It may help students to understand the categories of likelihood with an example of opening a book to a random page:

• Impossible: Opening a 100 page book to page -300.
• Unlikely: Opening a 100 page book to exactly page 45.
• Equally likely as not: Opening a 100 page book to a page numbered less than 51.
• Likely: Opening a 100 page book to a page numbered greater than 10.
• Certain: Opening a 100 page book to a page numbered less than 1,000.

Allow students 5--7 minutes quiet work time followed by partner and whole-class discussion.

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 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 agree on what goes in each category.

Questions for discussion

• "Were any of the scenarios listed difficult to categorize?"
• "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 engage in MP8 by recognizing a structure beginning to form with the results. Activities in later lessons will more formally show this structure forming from the repeated processes.

Define a chance experiment. A chance experiment is the process of making an observation of something when there is uncertainty about which of two or more possible outcomes will occur. Each time a standard number cube is thrown, it is a chance experiment. This is because we cannot be certain of which number will be the outcome of this experiment.

Some questions for discussion:

• "Who did you expect to win each time: the person choosing the number or the other person? Explain your reasoning."
• "In one round of the game it was more likely that the person choosing the numbers would win. We’ll be talking a lot in this unit about how likely or probable an event is to happen and even assigning numbers to the likelihood."
• "What if we had to assign numbers to the 'likelihood' of the person choosing the numbers winning for each round? For each part of the game, what percentage or fraction would you assign to the likelihood of the person who chose the numbers winning?" (For the first round of the game, there is a 50% or \(\frac{1}{2}\) chance for each person to win. For the second round of the game, the person choosing the numbers should have a 67% or \(\frac{4}{6}\) or \(\frac{2}{3}\) chance to win.)
• "What percentage or fraction would you assign to waiting for less than 10 minutes before your order is taken at the fast food restaurant from the previous task?" (Answers vary. 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 a fraction of customers who order their food within 10 minutes.)

The last activity in this lesson moves one more step closer to quantifying likelihood of scenarios by ordering them individually rather than into groups. Some of the scenarios have a numerical probability expressed as a percentage, some in decimal form, some as a fraction, and some do not have a numerical probability given (MP7). This gives students the opportunity to work with probabilities expressed as percentages, decimals, and fractions.

You will need the blackline master for this activity.

Arrange students in groups of 3. Distribute one copy of the blackline master for each group.

Tell students to take turns ordering the cards by beginning with one card, then adding in additional cards one at a time before, after, or between cards as needed.

Give students 5 minutes for group work followed by a whole-class discussion.

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 were the numerical values of the likelihoods written?" (Some were written as percentages, some as fractions, and some as decimal values.)
• "How did you compare them when there was a mix of percentages, fractions, and decimals?"
• "Some of the cards did not have a percentage, fraction, or decimal. How did you determine where those cards would go in the order?"