In this lesson you will work with different models for organizing outcomes of multiple
events when both one event and another event occur. Throughout this lesson, use these
questions to help focus your team’s discussion.
Is there more than one event?
Do both one event and other events occur?
Are the events independent?
5-65. THE DOUBLE SPIN
A giant wheel is divided into 5 equal sections labeled –2, –1, 0, 1, and 3. At the Double Spin,
players spin the wheel shown at right two times. The sum of their spins determines whether
they win. Explore using 5-65 Spinner eTool (CPM).
Work with your team to determine probabilities of different outcomes by answering the
questions below.
A.
B.
C.
D.
Make a list of the possible sums you could get.
Which sum do you think will be the most probable?
Create a probability table that shows all possible outcomes for the two spins.
If Tabitha could choose the winning sum for the Double Spin game, what sum would
you advise her to choose? What is the probability of her getting that sum with two
spins?
5-66. Scott’s job at Crazy Creations Ice Cream Shop is to design new ice cream flavors. The
company has just received some new ingredients and Scott wants to be sure to try all of the
possible combinations. He needs to choose one item from each category to create the new
flavor.
Base flavor
Vanilla
Chocolate
a.
Chunky mix-in
Fruit swirl
Hazelnuts
Sprinkles
Toffee bits
Apricot
Plum
Berry
Grape
Without talking with your teammates, list three different
combinations Scott could try. Make sure you use the word “and.” Then share
your combinations with your study team. How many different combinations did
you find? Do you think you found all of the possibilities?
b. Creating a list of all of the possibilities would take time and require a lot of
writing the same words over and over. Because there are more than two options,
a probability table is also challenging. An alternative is creating a probability
tree to show the different combinations. A probability tree, like the one started at
right and on the Lesson 5.2.5 Resource Page, shows the different possibilities
branching off each other. In this case, the two segments on the left show the base
flavors. Each different mix-in choice branches off of the base flavor, and each
fruit swirl branches off each mix-in choice. The first letter of each choice is used
to label this diagram.
i. The bold line in the diagram shows the combination vanilla, toffee bits
and plum swirl. Complete the probability tree to show all of the possible
combinations.
c. How many different flavor combinations are possible? Where do you look on the
diagram to count the number of complete combinations?
d. Use your probability tree to help you find the probability that Scott’s final
combination will include plum swirl.
e. What is the probability that his final combination will include hazelnuts?
5-67. Scott’s sister loves hazelnuts and Scott’s little brother loves grape.
A. Recall that events are favorable outcomes. List all of the outcomes in Scott’s sister’s
event. List all the outcomes in Scott’s little brother’s event.
B. Two events are mutually exclusive if they have no outcomes in common. Do Scott’s
sister and little brother have mutually exclusive events?
C. What would two mutually exclusive events in the Crazy Creations Ice-Cream Shop
be?
5-68. In a power outage, Rona has to reach into her closet in the dark to get dressed. She is
going to find one shirt and one pair of pants. She has three different pairs of pants hanging
there: one black, one brown, and one plaid. She also has two different shirts: one white and
one polka dot.
A. Draw a probability tree to organize the different outfit combinations Rona might
choose.
B. What is the probability that she will wear both a polka dot shirt and plaid pants?
C. What is the probability that she will not wear the black pants?
D. For what kinds of problems can you also make a probability table? If it is possible,
make a probability table for Rona’s outfits. Which way of representing the outcomes
do you like better?
E. Are the events polka dot and plaid mutually exclusive? Explain. .
F. Are the events polka dot and white mutually exclusive? Explain.
5-69. Represent all of the possible outcomes using a list, probability table, or probability
tree. Then find the indicated probability in each situation below.
A. You flip a coin three times in a row and get heads exactly twice.
B. You spin the two spinners at right and exactly one spinner lands on 4.
C. At the car rental agency, you will be given either a truck or a sedan. Each model
comes in four colors: green, black, white, or tan. If there is one vehicle of each color
for each model available, what is the probability you will get a green truck?
5-70. LEARNING LOG
In your Learning Log, describe the methods for organizing outcomes in a probability
situation that you have learned in the past few lessons, such as systematic lists, probability
tables and probability trees. Describe situations for which each tool is appropriate and any
advantages and disadvantages with using it. You may want to include an example from your
recent work to help you explain. Title this entry “Methods to Organize Probability
Outcomes” and label it with today’s date.
Probability Models for Multiple Events
To determine all possible outcomes for multiple events when both one event and the
other occur, there are several different models you can use to help organize the
information.
Consider spinning each spinner at right once.
If you use a plan or a pattern to find all of the outcomes in an event, you are making a
systematic list. For example, assume that you first spin B on spinner 1. Then, list all of
the possible outcomes on spinner 2. Next, assume that your first spin is W on spinner 1,
and complete the list.
A probability table can also organize information if there are exactly two events. The
possibilities for each event are listed on the sides of the table as shown, and the
combinations of outcomes are listed inside the table. In the example at right, the possible
outcomes for spinner 1 are listed on the left side, and the possible outcomes for spinner 2
are listed across the top. The possible outcomes of the two events are shown inside the
rectangle. In this table, the top and side are divided evenly because the outcomes are
equally likely. Inside the table you can see the possible combinations of outcomes.
A probability tree is another method for organizing information. The different
outcomes are organized at the end of branches of a tree. The first section has B and W at
the ends of two branches because there are two possible outcomes of spinner 1, namely
B and W. Then the ends of three more branches represent the possible outcomes of the
second spinner, R,G, and Y. These overall possible outcomes of the two events are
shown as the six branch ends.