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.
Launch
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.
Activity
None
Student Lesson in Spanish
Which game are you more likely to win? Explain your reasoning.
Game 1: You flip a coin and win if it lands showing heads.
Game 2: You roll a standard number cube and win if it lands showing a number that is divisible by 3.
Student Response
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Building on Student Thinking
Some students may have trouble comparing and . Ask students how they might write these values as decimals or to draw a shape and divide it into 6 equal regions, then think of what it would look like to shade half of the regions or of the regions.
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.
Activity Synthesis
Have partners share their answers and display the results for all to see. Select at least 1 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 2 ways to win with the standard number cube and only 1 way to win on the coin, the greater number of possible outcomes in the second game makes it less likely to provide a win.
3.2
Activity
Standards Alignment
Building On
Addressing
7.SP.C.7.a
Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
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).
This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas.
Launch
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 being 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 20, 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 and follow with a whole-class discussion.
Activity
None
For each chance experiment, list the sample space and tell how many outcomes there are.
Han rolls a standard number cube once.
Clare spins this spinner once.
A circular spinner divided into 4 equal parts. The top left section is red and labeled “R.” The top right section is blue and labeled “B.” The bottom right section is green and labeled “G.” The bottom left section is yellow and labeled “Y.” The spinner dial points to the section labeled "G."
Kiran selects a letter at random from the word “MATH.”
Mai selects a letter at random from the alphabet.
Noah picks a card at random from a stack that has cards numbered 5 through 20.
Next, compare the likelihood of these outcomes. Be prepared to explain your reasoning.
Is Clare more likely to have the spinner stop on the red or blue section?
Is Kiran or Mai more likely to get the letter T?
Is Han or Noah more likely to get a number that is greater than 5?
Suppose you have a spinner that is evenly divided showing all the days of the week. You also have a bag of papers that list the months of the year. Are you more likely to spin the current day of the week or pull out the paper with the current month?
Student Response
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Building on Student Thinking
Activity Synthesis
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)
Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of outcomes. Terms may include “random,” “outcome,” and “sample space.” Supports accessibility for: Conceptual Processing, Language
3.3
Activity
Standards Alignment
Building On
Addressing
7.SP.C.6
Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
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).
Launch
Arrange students in groups of 4. Provide each group with a paper bag containing 1 set of slips cut from the blackline master. Allow students 10 minutes for partner work and follow with a whole-class discussion.
Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, pause to check for understanding after 3–5 minutes of work time. Supports accessibility for: Organization, Attention
Activity
None
Your teacher will give your group a bag of paper slips with something printed on them. Repeat these steps until everyone in your group has had a turn.
As a group, guess what is printed on the papers in the bag and describe your guess in the table.
Without looking in the bag, one person takes out one of the papers and shows it to the group.
Everyone in the group writes what is printed on the paper.
The person who took out the paper puts it back into the bag, shakes the bag to mix up the papers, and passes the bag to the next person in the group.
Guess the
sample space.
What is printed
on the paper?
person 1
person 2
person 3
person 4
How is guessing the sample space the fourth time different from the first?
Without looking in the bag, what could you do to get a better guess of the sample space?
Look at all the papers in the bag. Are any of your guesses correct?
Are all of the possible outcomes equally likely? Explain.
Use the sample space to determine the probability that a fifth person would get the same outcome as person 1.
Student Response
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Building on Student Thinking
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?”
Activity Synthesis
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.)
MLR8 Discussion Supports. During group work, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “I heard you say . . . .” Original speakers can agree or clarify for their partner. Advances: Listening, Speaking
Lesson Synthesis
Consider asking some of the questions:
“If you choose one letter at random from the English alphabet, how many outcomes are there in the sample space? How many outcomes are in the event that a vowel (not including Y) is chosen?” (There are 26 outcomes in the sample space. There are 5 outcomes that are vowels: A, E, I, O, and U.)
“What is the sample space of a chance experiment? How is the number of outcomes in the sample space related to the probability of an event if the outcomes in the sample space are equally likely?” (The sample space is the list of possible outcomes for an experiment. The number of outcomes in the sample space influences the denominator of the probability when written as a fraction.)
“When there are 100 different outcomes in the sample space that are equally likely, what is the probability that a specific outcome will happen?” ( or 1% or 0.01)
Student Lesson Summary
The probability of an event is a measure of the likelihood that the event will occur. Probabilities are expressed using numbers from 0 to 1.
If the probability is 0, that means the event is impossible. For example, when a coin is flipped, the probability that it will turn into a bottle of ketchup is 0. The closer the probability of some event is to 0, the less likely it is.
If the probability is 1, that means the event is certain. For example, when a coin is flipped, the probability that it will land somewhere is 1. The closer the probability of some event is to 1, the more likely it is.
If we list all of the possible outcomes for a chance experiment, we get the sample space for that experiment. For example, the sample space for rolling a standard number cube includes six outcomes: 1, 2, 3, 4, 5, and 6. The probability that the number cube will land showing the number 4 is . In general, if all outcomes in an experiment are equally likely and there are possible outcomes, then the probability of a single outcome is .
Sometimes we have a set of possible outcomes and we want one of them to be selected at random. That means that we want to select an outcome in a way that is based on chance. For example, if two people both want to read the same book, we could flip a coin to see who gets to read the book first.
Standards Alignment
Building On
Addressing
7.SP.C.5
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
