Lesson 13

The purpose of this warm-up is for students to choose the more likely event based on their intuition about the possible outcomes of two chance experiments. The activities in this lesson that follow define probability and give ways to compute numerical values for the probability of chance events such as these.

Arrange students in groups of 2. Give students 1 minute of quiet work time followed by time to share their response with a partner. Follow with a whole-class discussion. 

Some students may have trouble comparing \(\frac{1}{2}\) and \(\frac{2}{6}\). Review how to compare fractions with these students.

Some students may struggle with the wording of the second game. Help them understand what it means for a number to be divisible by a certain number and consider providing them with a standard number cube to examine the possible values.

Have partners share their answers and display the results for all to see. Select at least one student for each answer provided to give a reason for their choice. 

If no student mentions it, explain that the number of possible outcomes that count as a win and the number of total possible outcomes are both important to determining the likelihood of an event. That is, although there are two ways to win with the standard number cube and only one way to win on the coin, the greater number of possible outcomes in the second game makes it less likely to provide a win.

Following the warm-up discussion in which the importance of the number of possible outcomes from an experiment is discussed, students are introduced to the term sample space. Students examine experiments to determine the set of outcomes in the sample space and then use the sample spaces to think about the likelihood of the events. Following the activity, students engage in MP6 by attaching numerical values to the likelihood of events through the word probability. 

In some cases, the actual sample space is unknown. For example, in the warm-up for the first lesson of this unit, all the different types of fish in the lake may not be known. In these cases, probabilities may be less precise, but can still be estimated based on the outcomes from previous experiments. Students will begin to explore this idea in the next activity and later lessons will explore the concept in more detail.

Arrange students in groups of 2.

Define random as doing something so that the outcomes are based on chance. An example is putting the integers 1 through 20 on a spinner with each number in an equal sized section. Something that is not random might be answering a multiple choice question on a test for a subject you've studied.

Explain that, for a chance experiment, each possible result is called an outcome. The set of all possible outcomes is called the sample space. Spinning a spinner with equal sized sections marked 1 through 20 has a possible outcome of 8, but neither heads nor green is a possible outcome. The sample space is made up of all integers from 1 through 20.

Give students 10 minutes of partner work time followed by whole-class discussion.

Note that all of the outcomes are equally likely within each sample space. This is not always the case, but it is in these examples.

Explain that sometimes it is important to have an actual numerical value rather than a vague sense of likelihood. To answer how probable something is to happen, we assign a probability.

Probabilities are values between 0 and 1 and can be expressed as a fraction, decimal, or percentage. Something that has a 50% chance of happening, like a coin landing heads up, can also be described by saying, “The probability of a coin landing heads up is \(\frac{1}{2}\).” or “The probability of the coin landing heads up is 0.5.”

When each outcome in the sample space is equally likely, we may calculate the probability of a desired event by dividing the number of outcomes for which the event occurs by the total number of outcomes in the sample space. 

• "A standard number cube is rolled. What is the sample space?" (1, 2, 3, 4, 5, 6) 
  • "How many outcomes are in the sample space?" (6)
• "What is the probability of rolling a 3? Explain your reasoning." (\(\frac{1}{6}\) since there is a single 3 and 6 outcomes in the sample space.)
• "An experiment has one of each different possible outcome. The probability of getting one of the outcomes is \(\frac{1}{30}\). How many outcomes are in the sample space?" (30)

In this activity, students are introduced to the idea that not all sample spaces are obvious before actually doing the experiment. Therefore, it is not always possible to calculate the exact probabilities for events before doing or simulating the experiment. In such situations, it is important to reason abstractly about the scenario (MP2) to gain an understanding of the situation. Students refine their guesses about the sample space by repeatedly drawing items from a bag and looking for patterns in this repetition (MP8). In later lessons, students learn how to estimate probabilities from simulations. 

You will need the blackline master for this activity.

Arrange students in groups of 4. Provide each group with a paper bag containing 1 set of slips cut from the blackline master. 10 minutes partner work followed by a whole-class discussion.

Students may think that the phrase "equally likely" means there is a 50% chance of it happening. Tell students that, in this context, each outcome is equally likely if the probability does not change if you change the question to a different outcome in the sample space. For example, "What is the probability you get a letter A from this bag?" has the same answer as the question, "What is the probability you get a letter B from this bag?"

The purpose of this discussion is for students to understand that often, in the real-world, we do not know the entire sample space before doing the experiment. They will learn in later lessons how to estimate the probabilities for such experiments.

Consider asking some of the following questions:

• "After the first paper is drawn, a group guesses, 'A bunch of letter Cs.' What might they have picked on their first paper that would lead to that guess? What could that group get on their second paper that would make them change their guess? Could they get something for the second paper that would make them sure their guess was right?" (They probably picked a letter C on their first draw. Any other letter on the second draw should make them change their guess. No, they might get another C that would make them think they are right, but with only two tries, there is still a good chance that other letters are in the bag.)
• "After the second paper is drawn, a group guesses, 'All of the consonants.' What might they have picked in their first two papers that would lead to that guess? What could that group get on their third paper that would make them change their guess? Could they get something that would make them more sure of their guess?" (They probably picked two consonants on their first two draws. If they picked a vowel, they would have to change their guess. If they picked another consonant, they might feel better about the guess, but should still not be certain of it.)
• "How did you refine your predictions with each round?"
• "If you had a new bag of papers and you took out papers 50 times and never got a 'Z,' would that mean there is no 'Z' in the bag?" (Not necessarily, but it might make me wonder if it's not in there.)