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Lecture Slides
Elementary Statistics
Eleventh Edition
and the Triola Statistics Series
by Mario F. Triola
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Chapter 4
Probability
4-1 Review and Preview
4-2 Basic Concepts of Probability
4-3 Addition Rule
4-4 Multiplication Rule: Basics
4-5 Multiplication Rule: Complements and
Conditional Probability
4-6 Probabilities Through Simulations
4-7 Counting
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Section 4-7
Counting
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Key Concept
In many probability problems, the big obstacle
is finding the total number of outcomes, and
this section presents several methods for
finding such numbers without directly listing
and counting the possibilities.
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Fundamental Counting Rule
For a sequence of two events in which
the first event can occur m ways and
the second event can occur n ways,
the events together can occur a total of
m  n ways.
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Test Question
Imagine that you are taking a test. The first
question is True/False, the second is a multiple
choice question with four possible answers, and
the third question is multiple choice with 6
possible answers.
If you didn’t study, and must select answers to the
three questions totally at random, what is the
probability of getting them all correct?
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Notation
The factorial symbol ! denotes the product of
decreasing positive whole numbers.
For example,
4!  4  3  2  1  24
By special definition, 0! = 1.
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Factorials
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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 in n
different ways, the second item may be
selected in n – 1 ways, and so on.)
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Permutations Rule
(when items are all different)
Requirements:
1. There are n different items available. (This rule does not
apply if some of the items are identical to others.)
2. We select r of the n items (without replacement).
3. We consider rearrangements of the same items to be
different sequences. (The permutation of ABC is different
from CBA and is counted separately.)
If the preceding requirements are satisfied, the number of
permutations (or sequences) of r items selected from n
available items (without replacement) is
n!
n Pr 
(n  r )!
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Permutations
• Evaluate the expression: 5P3
n!
n Pr 
(n  r )!
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Permutations Rule
(when some items are identical to others)
Requirements:
1. There are n items available, and some items are identical to
others.
2. We select all of the n items (without replacement).
3. We consider rearrangements of distinct items to be different
sequences.
If the preceding requirements are satisfied, and if there are n1
alike, n2 alike, . . . nk alike, the number of permutations (or
sequences) of all items selected without replacement is
n!
n1 !.n2 !........nk !
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Permutation Rule
• In this classroom, there are 5 males and 4
females. How many different ways are there
for the nine of us to line up at the door when
the bell rings?
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Combinations Rule
Requirements:
1. There are n different items available.
2. We select r of the n items (without replacement).
3. We consider rearrangements of the same items to be the
same. (The combination of ABC is the same as CBA.)
If the preceding requirements are satisfied, the number of
combinations of r items selected from n different items is
n!
n Cr 
(n  r )!r !
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Combination Problem
• The eight of you decide to start the Statistics
Club at RHS. How many way are there to
select a President, Vice-President, and
Treasurer?
• How many ways are there to select a three
member recruitment committee?
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Permutations versus
Combinations
When different orderings of the same
items are to be counted separately, we
have a permutation problem, but when
different orderings are not to be counted
separately, we have a combination
problem.
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Recap
In this section we have discussed:
 The fundamental counting rule.
 The factorial rule.
 The permutations rule (when items are all
different).
 The permutations rule (when some items
are identical to others).
 The combinations rule.
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