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Chapter 5: Elementary Probability Theory
Section 5.3: Trees and Counting Techniques
Name
Date
Objective:
•
Organize outcomes in a sample space using tree diagrams.
•
Compute the number of ordered arrangements of outcomes using permutations.
•
Computer the number of (non-ordered) groupings of outcomes using permutations.
This worksheet will walk you through the steps to determine the number of outcomes of an experiment
using a tree diagram or a rule/formula for permutations or combinations.
Sometimes, you will be told which method to use to determine the number of outcomes of an
experiment. If not, use the following procedure to decide your approach:
A.
Use a tree diagram to determine the number of outcomes for the scenario in Exercise 6 in your
textbook, which is reproduced here:
Step 1:
Draw the tree and label the branches. Count the possible outcomes.
This step ensures that you address each possible outcome.
Chapter 5: Elementary Probability Theory
Section 5.3: Trees and Counting Techniques
Example:
Draw a tree diagram that shows all of the possible schedules for the scenario in Exercise 9 in
your textbook, which is reproduced here:
Label the
starting point.
The numbered events are the section numbers she has to choose from. There
are 2 for psychology and A and P, and three for Spanish II.
There are 12
possible outcomes.
Chapter 5: Elementary Probability Theory
Section 5.3: Trees and Counting Techniques
Instructions:
Complete the tree diagram below using the information in Exercise 6 in your Introductory
Statistics textbook. Label the events at the end of each branch. Count and state the number of
outcomes.
Start
Coin Flip Result
Die Toss Result
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Chapter 5: Elementary Probability Theory
Section 5.3: Trees and Counting Techniques
Step 2:
List the outcomes. (Note: Depending on the exact information you are asked for, this step may
be optional.)
Sometimes, you also want to know exactly what the possible outcomes are. You can make a comma-
separated list or table to list the outcomes. To get the outcomes, read along each branch of the tree to
the end, and list the node labels.
Example:
Using a table, list all the possible schedules for the scenario in Exercise 9 in which Jacqueline
takes Psychology section 1.
Instructions:
Complete the table below with the possible coin flip/die toss sequences from Exercise 8 in
your textbook. Then answer parts (b) and (c) of Exercise 8.
Possible Coin Flip / Die Toss Outcomes
Coin Flip Result
Die Toss Result
Outcome
H
1
H, 1
The first complete branch of the tree becomes the first row of the table.
8(b):
8(c):
Chapter 5: Elementary Probability Theory
Section 5.3: Trees and Counting Techniques
B.
Use a counting method to determine the number of outcomes for the scenario in Exercise 23 in
your textbook, which is reproduced here:
As indicated in the Procedure on Page 1 of this worksheet, first determine the types of outcomes from
the experiment: a series of stages with various outcomes (use the multiplication rule or tree diagram);
ordered subgroups of
r
items taken from a group of
n
items (use the permutation rule); or nonordered
subgroups of
r
items taken from a group of
n
items (use the combination rule).
Example:
Determine the number of seating arrangements for five people in ten chairs.
Step 1:
Determine which counting method or rule to use.
Here, two different orderings of the same five people are considered different seating arrangements.
Order matters
, so use the permutation rule:
(
)
,
!
!
n r
n
P
n
r
=
−
Step 2:
Substitute the values of
n
and
r
into the equation and simplify.
The larger group is the number of chairs, so
n
= 10. The subgroup is the number of people, so
r
= 5.
(
)
10,5
10!
10!
3,628,800
30,240
10
5 !
5!
120
P
=
=
=
=
−
Instructions:
Select the correct counting method or rule and then use it to complete Exercise 23 in your
textbook.
There are 10 chairs.
There are 5 people to arrange in the chairs.
There are 30,240
possible arrangements.
Chapter 5: Elementary Probability Theory
Section 5.3: Trees and Counting Techniques
C.
Instructions:
Use a counting method determine the number of outcomes for the scenario in Exercise
24 in your textbook, which is reproduced here:
As indicated in the Procedure on Page 1 of this worksheet, you want to first see which of the following
types of outcomes you have: a series of stages with various outcomes (use the multiplication rule or tree
diagram); ordered subgroups of
r
items taken from a group of
n
items (use the permutation rule); or
nonordered subgroups of
r
items taken from a group of
n
items (use the combination rule). That will
help you figure out which counting rule to use.
Example:
Determine the number of different committees of 4 people can be selected from a group of 12
faculty members.
Step 1:
Determine which counting rule to use.
In this case, the people are selected for a committee, which is a nonordered group.
Order does not
matter
,
so use the combination rule:
(
)
,
!
!
!
n r
n
C
r
n
r
=
−
Step 2:
Substitute the values of
n
and
r
into the equation and simplify.
You are selecting from a group of 12 faculty members, so
n
= 12. The subgroup is the number of people
on the committee, so
r
= 4.
(
)
12,4
12!
12!
495
4! 12
4 !
4!8!
C
=
=
=
−
There are 12 faculty members.
There will be 4 people on the committee.
There are 495
different committees
possible.
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Chapter 5: Elementary Probability Theory
Section 5.3: Trees and Counting Techniques
Instructions:
Select the correct counting method or rule and then use it to complete Exercise 24 in your
textbook.