Probability As A Measure Of Likelihood
Students will begin to think about probability by considering how likely it is that their house will be struck by lightning. They will consider the relative likelihood of familiar events (e.g., outdoor temperature, test scores) on the continuum between impossible and certain. Students will discuss where on the continuum "likely," "unlikely," and "equally likely as unlikely" are.
As students begin their study of probability, they look at the likelihood of events. Students have an intuitive sense of likelihood, even if no numbers or ratios are attached to the events. For example, there is clearly a better chance that a specific student will be chosen at random from a class than from the entire school.
Goals and Learning Objectives
- Think about the concept of likelihood.
- Understand that probability is a measure of likelihood.
- Informally estimate the likelihood of certain events.
- Begin to think about why one event is more likely than another.
SWD: Students with disabilities may need additional support seeing the relationships among problems and strategies. Throughout this unit, keep anchor charts available and visible to assist them in making connections and working toward mastery. Provide explicit think alouds comparing strategies and making connections. In addition, ask probing questions to get students to articulate how a peer solved the problem or how one strategy or visual representation is connected or related to another.
Have students look at the image of lightning and read the question, "How likely do you think it is that your house will be struck by lightning this year?".Have them talk to their partner and then share their responses with the class. Briefly discuss the responses.
During the discussion, ask:
- How did you decide how likely it is that your house would be struck by lightning?
- Can you think of other events that would be about as likely to happen?
ELL: Be sure that your pace is appropriate when posing these questions, especially when interacting with ELLs.
Think about this question, and then discuss your ideas with a partner.
- How likely do you think it is that your house will be struck by lightning this year?
Discuss the Math Mission. Students will think about what makes one event more likely to occur than another event.
ELL: Identifying key words are crucial for students. Spend some time discussing the key vocabulary in this unit.
Think about what makes one event more likely to occur than another event.
Struck by Lightning
Have students mark the Likelihood Line interactive for the lightning question. Conduct an informal class poll, reminding students of mean and outliers if necessary.
Briefly discuss events that are impossible (e.g., a unicorn will walk into the room) or certain (e.g., the sun will come up tomorrow). (It is interesting to discuss what certain means to students.) Tell students that most events are neither impossible nor certain, but that they can be very unlikely or very likely. If we don't agree that the sun is certain to come up tomorrow, we can agree that it is very likely.
Briefly discuss events that are equally likely as unlikely (students are likely to suggest flipping a coin).
SWD: Familiarize students with the technology tools and ensure students can access the tools on the tablet successfully. Check the settings, adjust for easier usability for all students and customize the display of information.
- Answers will vary. Students should recognize that this event is very unlikely and place it close to impossible (but not at impossible).
- Answers will vary. As mentioned in the lesson guide, a possible example is a unicorn walking into the classroom.
- Answers will vary. As mentioned in the lesson guide, a possible example is the sun rising.
- Answers will vary. Possible answers: flipping tails/heads on a coin, picking a black/red card from a standard 52 card deck, rolling an even/odd number on a 6-sided die.
Struck by Lightning
Use the Likelihood Line interactive to help you think about likelihood.
- Where on the line would you place the likelihood of your house being struck by lightning? Put a mark at that point.
- Can you think of an event that is impossible—that cannot happen? Write an example of one.
- Can you think of an event that is certain? Write an example.
- Are there events you can think of that are equally likely to happen as to not happen? Write an example.
INTERACTIVE: Likelihood Line
Tell students that if an event is labeled to the right of center, we would say that it is likely; if it is labeled to the left, we would say that it is unlikely. The closer an event is to either extreme (Impossible or Certain), the more likely/unlikely it is happen.
Have students work quickly through the likelihood problems using the Likelihood Statements interactive. The discussion will occur in the Ways of Thinking portion of the lesson.
The problem begins with events that are easier to think about based on experience and more difficult to quantify with data. The last few events encourage students to informally think about the ratios involved to determine likelihood.
Most of the events will have a fairly narrow range of guesses. However, the ones about individual students (sending a text, buying lunch, getting a high score) could vary widely. Keep an eye on the range of guesses for all questions.
If students have difficulty understanding where to mark the line, ask:
- Do you think the event is closer to impossible or certain?
- Do you think the event is likely (closer to center) or very likely (closer to certain)?
- Do you think the event is unlikely (closer to center) or very unlikely (closer to impossible)?
Look for the following in student responses:
- Events with a large range of responses
- Events with a small range of responses
- Events with a typical response of very unlikely, very likely, or equally likely as unlikely
Mathematical Practice 4: Model with mathematics.
- Students begin to build their intuition about probability by considering real-world situations. Subsequent lessons will introduce the mathematical models that can be used to represent these and other situations.
- Answers will vary as described in the Mathematics section.
For each event, use the Likelihood Statements interactive to mark the line with the letter of the event to indicate its likelihood.
What is the likelihood that...
a. It will rain tomorrow?
b. Someone will be absent?
c. The temperature will be above 60°F?
d. You will send a text sometime today?
e. Your teacher will assign homework?
f. You will buy lunch at school?
g. You will get a high score on the next math test?
h. Out of all the students at your school, you will be chosen at random to win a prize?
i. Out of all the students at your school, you will not be chosen at random to win a prize?
j. Out of all the students in your class, you will be chosen at random to win a prize?
k. Out of all the students in your class, a girl will be chosen at random to win a prize?
INTERACTIVE: Likelihood Statements
Likelihood Both Will Happen
Look for students who predict that the likelihood of two events occurring is lower than the likelihood of each event individually. Also look for students who predict a higher likelihood.
- Answers will vary. In general, the likelihood of any two given events both happening should be lower than the likelihood of either individual event happening. Special cases include:
- If a student picks mutually exclusive events (e.g., h and i), it is impossible for both to occur.
- If a student reasons either of the two events to be impossible, then the likelihood for both events must also be impossible—regardless of the likelihood of the other event.
- If a student reasons both events to be certain independently, they must also be certain for both to happen.
Likelihood Both Will Happen
Choose two events from the list. What is the likelihood they will both happen? Use the Two Events Likelihood Line interactive to mark the likelihood on the line.
a. It will rain tomorrow.
b. Someone will be absent.
c. The temperature will be above 60°F.
d. You will send a text sometime today.
e. Your teacher will assign homework.
f. You will buy lunch at school.
g. You will get a high score on the next math test.
h. Out of all the students at your school, you will be chosen at random to win a prize.
i. Out of all the students at your school, you will not be chosen at random to win a prize.
j. Out of all the students in your class, you will be chosen at random to win a prize.
k. Out of all the students in your class, a girl will be chosen at random to win a prize.
INTERACTIVE: Two Events Likelihood Line
This will be a teacher-led discussion. Discuss the results of students' responses.
Ask the following:
- What statement about likelihood could you make for each event (e.g., it is very unlikely that I would be chosen at random for a prize)?
- How did you decide where to place your mark for each event?
- Why did the class seem to agree on some events and not others?
- We agree that some of the events are unlikely, but how do you know that one is more unlikely than the other?
- How do the events “you will be chosen to win a prize” and “you will not be chosen to win a prize” go together?
- How would the likelihood compare for the opposite event for each event in the list—e.g., "It will not rain tomorrow”?
Talk about students' responses to the likelihood of two events. Mention to students that these are examples of compound events, which will be discussed later in the unit.
Although numbers and ratios are discussed only informally, students are beginning to think about probability and complementary events.
SWD: Revisit the vocabulary introduced in this lesson. As students present their solutions to the Work Time problems, make note of key words and write them on a chart. Provide plenty of repetition and review of new terminology. Make sure all students have these terms in their notebook.
ELL: Give students plenty of wait time as some of them are learning and processing new mathematical information in a second language. Use a chart to show a visual representation of the likelihood line with a continuum of where likely, unlikely, and equally likely as unlikely are located with respect to each other.
Ways of Thinking: Make Connections
Take notes about how your responses compared to those of your classmates.
As your classmates present, ask questions such as:
- What statement about likelihood could you make for each event? (For example: It is very unlikely that I would be chosen at random for a prize.)
- How did you decide where to place a mark for each event?
- What is the likelihood that both events, "You will be chosen at random to win a prize" and "You will not be chosen to win a prize," will happen?
- How does the likelihood of the opposite of each event compare to the likelihood of the original event? (For example, “It will not rain tomorrow” versus “It will rain tomorrow” and “The temperature will be below 60°F” versus “The temperature will be above 60ºF”)?
Reflect on Your Work
Have each student write a brief reflection before the end of the class. Review the reflections to find out what students consider to be likely and unlikely events.
Write a reflection about the ideas discussed in class today. Use the prompts below if you find it to be helpful.
In the next 24 hours, here is a list of some things that are likely to occur…
In the next 24 hours, here is a list of some things that are unlikely to occur...