Section 12.5
Tree Diagrams
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
INB Table of Contents
Date
2.3-2
Topic
Page #
October 2, 2013
Section 12.5 Examples
34
October 2, 2013
Section 12.5 Notes
35
October 2, 2013
Section 12.6 Examples
36
October 2, 2013
Section 12.6 Notes
37
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
 Counting Principle
 Tree Diagrams
12.5-3
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Counting Principle
If a first experiment can be performed in M distinct ways
and a second experiment can be performed in N distinct
ways, then the two experiments in that specific order can
be performed in M • N distinct ways.
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Definitions

Sample space: A list of all possible outcomes
of an experiment.

Sample point: Each individual outcome in the
sample space.

Tree diagrams are helpful in determining
sample spaces.
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Example 1: Selecting Balls
without Replacement
Two balls are to be selected without replacement from a
bag that contains one red, one blue, one green and one
orange ball.
a)
Use the counting principle to determine the number of
points in the sample space.
b)
Construct a tree diagram and list the sample space.
c)
Find the probability that one orange ball is selected.
d)
Find the probability that a green ball followed by a red
ball is selected.
12.5-6
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12.5-9
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Example 3: Selecting Ticket
Winners
A radio station has two tickets to give away to a Bon Jovi concert. It
held a contest and narrowed the possible recipients down to four
people: Christine (C), Mike Hammer (MH), Mike Levine (ML), and
Phyllis (P). The names of two of these four people will be selected at
random from a hat, and the two people selected will be awarded the
tickets.
a)
Use the counting principle to determine the number of points in the
sample space.
b)
Construct a tree diagram and list the sample space.
c)
Determine the probability that Christine is selected.
d)
Determine the probability that neither Mike Hammer nor Mike
Levine is selected.
e)
Determine the probability that at least one Mike is selected.
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b) Construct a tree diagram and list
the sample space.
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P(event happening at least once)
 event happening
 event does 
P
1 P


 at least once

 not happen
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P(event happening at least once)
In part d) of Example 3, we found that
1
P(neither Mike is selected) 
6
In part e) we could have used
 at least one Mike
 neither Mike
P

1

P

 is selected 
 is selected
1 5
P(at least one Mike selected)  1  
6 6
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Section 12.6
OR and AND
Problems
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What You Will Learn

Compound Probability

OR Problems

AND Problems

Independent Events
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Compound Probability
In this section, we learn how to solve
compound probability problems that
contain the words and or or without
constructing a sample space.
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OR Probability
The or probability problem requires
obtaining a “successful” outcome for at
least one of the given events.
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Probability of A or B
To determine the probability of A or B,
use the following formula.
P(A or B)  P(A)  P(B)  P(A and B)
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Example 1: Using the Addition
Formula
Each of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 is
written on a separate piece of paper. The 10 pieces of
paper are then placed in a hat, and one piece is randomly
selected. Determine the probability that the piece of paper
selected contains an even number or a number greater
than 6.
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Mutually Exclusive

Two events A and B are mutually
exclusive if it is impossible for both
events to occur simultaneously.

If two events are mutually exclusive, then
the P(A and B) = 0.

The addition formula simplifies to
P(A or B)  P(A)  P(B).
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Example 3: Probability of A or B
One card is selected from a standard deck of playing
cards. Determine whether the following pairs of events are
mutually exclusive and determine
P (A or B).
a)
A = an ace, B = a 9
b)
A = an ace, B = a heart
c)
A = a red card, B = a black card
d)
A = a picture card, B = a red card
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And Problems
The and probability problem
requires obtaining a favorable outcome
in each of the given events.
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Probability of A and B
To determine the probability of A and
B, use the following formula.
P(A and B)  P(A)  P(B)
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Probability of A and B
Since we multiply to find P (A and B), this formula
is sometimes referred to as the multiplication
formula.
When using the multiplication formula, we always
assume that event A has occurred when
calculating P(B) because we are determining the
probability of obtaining a favorable outcome in
both of the given events.
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Example 5: An Experiment
without Replacement
Two cards are to be selected without replacement from a
deck of cards. Determine the probability that two spades
will be selected.
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Independent Events
Event A and event B are independent
events if the occurrence of either
event in no way affects the probability
of occurrence of the other event.
Rolling dice and tossing coins are
examples of independent events.
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Example 6: Independent or
Dependent Events?
One hundred people attended a charity benefit to raise
money for cancer research. Three people in attendance
will be selected at random without replacement, and each
will be awarded one door prize. Are the events of selecting
the three people who will be awarded the door prize
independent or dependent events?
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Independent or Dependent
Events?
In general, in any experiment in which
two or more items are selected without
replacement, the events will be
dependent.
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