Unit 8, Lesson 16
Multi-step Experiments
Accelerated 6
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The purpose of this Warm-up is to get students thinking about probabilities and spinners, which will be useful when students compute these values in a later activity. While students may notice and wonder many things about these spinners, the probabilities are the important discussion points.
Arrange students in groups of 2. Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner.
What do you notice? What do you wonder?
Circular spinner divided into four equal parts. The first part is red and labeled “R,” the second part is blue and labeled “B,” the third part is green and labeled “G,” and the fourth part is yellow and labeled “Y.” The pointer is in the part labeled “B.”
Circular spinner divided into five equal parts. Starting from the top right, and moving clockwise, the first part is labeled 1, the second, 2, the third, 3, the fourth, 4, and the fifth, 5. The pointer is in the part labeled “5.”
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If probability does not come up during the conversation, ask students to discuss this idea.
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In this activity, students are reminded how to calculate probability based on the number of outcomes in the sample space, then apply that to multi-step experiments. The events are described in everyday language, so students need to reason abstractly (MP2) to identify the outcomes described. This lesson begins with students returning to a problem they have previously seen when writing out the sample space. This will save students some time if they can recall or refer back to the initial problem.
Tell students: “For sample spaces where each outcome is equally likely, recall that the probability of an event can be computed by counting the number of outcomes in the event and dividing that number by the total number of outcomes in the sample space.”
For example, there are 12 possible outcomes when flipping a coin and rolling a number cube. If we want the probability of getting heads and rolling an even number, we count that there are 3 ways to do this (H2, H4, and H6) out of the 12 outcomes in the sample space. So the probability of getting heads and an even number should be (
Remind students that they have already drawn out the sample space for this chance experiment in a previous activity, and they may use that to help answer the questions.
Give students 5 minutes quiet work time followed by partner and whole-class discussion.
What is the probability of getting:
Circular spinner divided into four equal parts. The first part is red and labeled “R,” the second part is blue and labeled “B,” the third part is green and labeled “G,” and the fourth part is yellow and labeled “Y.” The pointer is in the part labeled “B.”
Circular spinner divided into five equal parts. Starting from the top right, and moving clockwise, the first part is labeled 1, the second, 2, the third, 3, the fourth, 4, and the fifth, 5. The pointer is in the part labeled “5.”
The purpose of this discussion is for students to explain their interpretations of the questions and share methods for solving.
Some questions for discussion:
Poll the Class
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In this activity, students continue to compute probabilities for multi-step experiments using the number of outcomes in the sample space. They are assigned either a list, table, or tree to examine the sample space for a chance experiment, then they use their understanding of the sample space to write probabilities for various events happening. Students are also reminded that some events have a probability of 0, which represents an event that is impossible. In the discussion following the activity, students are asked to think about the probabilities of two events that make up the entire sample space and have no outcomes common to both events (MP2).
Keep students in groups of 2.
Assign each group a representation for writing out the sample space: a tree, a table, or a list. Tell students that they should write out the sample space for the first problem using the representation they were assigned. (This was done for them in a previous lesson, and they are allowed to use those as a guide if they wish.)
Tell students that they should work on the first problem only and then pause for a discussion before proceeding to the next problems.
Give students 2 minutes of partner work time for the first problem followed by a pause for a whole-class discussion centered around the different representations for sample space.
After all groups have completed the first question, select at least one group for each representation, and have them explain how they arrived at their answer. As the groups explain, display the appropriate representations for all to see. Ask each of the groups how they counted the number of outcomes in the sample space as well as the number of outcomes in the event using their representation.
List:
Heads 1, heads 2, heads 3, heads 4, heads 5, heads 6, tails 1, tails 2, tails 3, tails 4, tails 5, tails 6
Table:
| 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|
| H1 | H2 | H3 | H4 | H5 | H6 |
| T1 | T2 | T3 | T4 | T5 | T6 |
Tree:
After students have had a chance to explain how they used the representations, ask students to give some pros and cons for using each of the representations. For example, the list method may be easy to write out and interpret but could be very long and is not the easiest method for keeping track of which outcomes have been written and which still need to be included.
Allow the groups to continue with the remaining problems, telling them they may use any method they choose to work with the sample space for these problems. Give students 10 minutes of partner work time followed by a whole-class discussion about the activity as a whole.
The purpose of the discussion is for students to explain their methods for solving the problems and to discuss how writing out the sample space aided in their solutions.
Poll the class on how they computed the number of outcomes in the sample space and the number of outcomes in the event for the second set of questions given these options: list, table, tree, computed outcomes without writing them all out, or another method.
Consider these questions for discussion:
Suppose we have two bags. One contains 1 star block and 4 moon blocks. The other contains 3 star blocks and 1 moon block.
If we select 1 block at random from each, what is the probability that we will get 2 star blocks or 2 moon blocks?
To answer this question, we can draw a tree diagram to see all of the possible outcomes.
A tree diagram. The first choice has 5 branches, representing the 5 blocks in the bag: one branch is labeled “star,” the other 4 are labeled “moon.” Each of these branches has 4 branches, representing the 4 blocks in the second bag. 3 branches are labeled “star” and one is labeled “moon.” The word “star” in the first choice, and the 3 “star” choices branching from it are highlighted gold. From the first choice, the word “moon” is highlighted blue on each of the four remaining branches. From each of those branches, the one choice of “moon” for each is also highlighted blue.
There are
In general, if all outcomes in an experiment are equally likely, then the probability of an event is the fraction of outcomes in the sample space for which the event occurs.
Help us improve by sharing suggestions or reporting issues.
Your teacher will assign you to use either a list, table, or tree. Be prepared to explain your reasoning.
A number cube is rolled and a coin is flipped.
What is the probability of getting heads and an odd number?
Pause here so your teacher can review your work.
You may use any method you wish to answer these questions. Suppose you roll two number cubes. What is the probability of getting:
Both cubes showing the same number?
Exactly one cube showing an even number?
At least one cube showing an even number?
Two values that have a sum of 8?
Some students may not recognize that rolling a 2 then a 3 is different from rolling a 3 then a 2. Ask students to imagine the number cubes are different colors to help see that there are actually 2 different ways to get these results.
Similarly, some students may think that HHT counts the same as HTH and THH. Ask the student to think about the coins being flipped one at a time rather than all tossed at once. Drawing an entire tree and seeing all the branches may further help.
Students may misread the problem and think that they replace the card before picking the next one. Ask these students to read the problem more carefully and ask them, “What is possible to get when you draw the second card while you already have a red card in your hand?”