Section 4.3 Trees and
Counting Techniques
Notes: Monday, Nov. 25
• Organize outcomes in a sample
space using tree diagrams.
• Define and use factorials.
• Explain how counting techniques
relate to probability in everyday
life
Multiplication Rule of Counting
# 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝒇𝒂𝒗𝒐𝒓𝒂𝒃𝒍𝒆 𝑡𝑜 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝐴
𝑃 𝐴 =
# 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒
In the last section, the
# of outcomes in a
sample space were
small or simple to
calculate.
In this section, we will be
able to count the number
of possible outcomes in
larger sample spaces or
ones formed by more
complicated events.
Multiplication Rule of Counting
• For a sequence of two events in which the first event can occur m
ways and the second events can occur n ways, the events together
can occur a total of 𝑚 ∗ 𝑛 ways.
Example: You have discovered that all of the gold in Fort Knox is
accessible with a two-character code consisting of a letter followed by a
digit. How many possible codes are there?
Example Problems:
Mult. Rule of Counting – More than 2 Events
1. Computer Design. In designing a computer, if a byte is defined to be a
sequence of 8 bits and each bit must be a 0 or 1, how many different bytes are
possible?
2. Chance and Skill. Bob claims that he has the ability to roll a die in such a way
that 6 will almost always occur. You test him by giving him a fair die, which he
proceeds to roll five times, getting a 6 each time. If Bob has no control over the
die, how many outcomes are possible with five rolls of a die? If Bob does get
five 6s in five rolls, does it appear that he has control of the die?
Example Problems: Mult. Rule of Counting & Tree Diagram
3. Jacqueline is in a nursing program and is required to take a course
in psychology and one in physiology (A and P) next semester. She
also wants to take Spanish II. If there are two section of psychology,
two of A and P, and three of Spanish II, how many different class
schedules can Jacqueline choose from? (Assume that the times of
the sections do not conflict.)
Table For Schedules Utilizing Section 1 of Psychology:
Tree Diagram For Selecting Class Schedules:
Psych
Section
A and P
Section
Spanish II
Section
Example: Tree Diagram & Probability
• Suppose there are 5 balls in a
bin. 3 balls are red, 2 are blue.
You are to draw out one ball,
note its color and set it aside.
Continue drawing balls, noting
the color and setting them
aside. What are the outcomes
of the experiment? What is the
probability of each outcome?
Example Problems: Mult. Rule of Counting & Factorials
• Survey Questions. When designing surveys, pollsters sometimes try
to minimize a lead-in effect by rearranging the order in which the
questions are presented. If Gallup plans to conduct a consumer
survey by asking subjects 5 questions, how many different versions of
the survey are required if all possible arrangements are included?
Factorial Symbol: !
Denotes the product of decreasing positive whole numbers.
By special definition, 0! = 1.
Factorial on TI-83 Plus:
Enter the number, press
MATH and select PRB and
menu item 4.
Factorial Rule
• A collection of n different items can be arranged in order n! different
ways. (This factorial rule reflects the fact that the first item may be
selected n different ways, the second item may be selected n – 1
ways, and so on.)
Routing (shortest possible routes):
Example Problem: Factorial Rule
• Air Routes. You have just started your own airline company called Air
America (motto: “Where your probability of a safe flight is greater
than zero”). You have one plane for a route connecting Austin, Boise,
and Chicago. How many routes are possible?
Homework
A#4.31
Due Tuesday, November 26
Page 160
Numbers 5 – 7 and 9 – 12