In this activity, 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. Students reason quantitatively about each situation and represent the likelihood using numbers and the word probability (MP2).

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to the last question. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.

If time allows, display these prompts for feedback: 

• “How many outcomes are in each event?”
• “How did you use the number of outcomes to determine the likelihood?”
• “How do you know . . . ? What else do you know is true?”

Close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. As time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.
After Stronger and Clearer Each Time, 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.

The probability of an event is a number that tells how likely it is to happen. 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 ” 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. 

• "What is the probability of spinning the current day from the last question? Explain your reasoning.” (, because today is 1 day out of the 7 possible days of the week)
• “What is the probability that Noah gets a number greater than 5? Explain your reasoning.” (, because there are 15 outcomes in the event and 16 outcomes in the sample space)

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. Students refine their guesses about the sample space by repeatedly drawing items from a bag and looking for patterns in this repetition (MP8).

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 that the sample space is '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.)