Transcript Chapter 9

Chapter
9
Probability
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All rights reserved
NCTM Standard: Data Analysis and
Probability
 K–2: Children should discuss events related to
their experience as likely or unlikely. (p. 400)
 3–5: Children should be able to “describe
events as likely or unlikely and discuss the
degree of likelihood using words such as
certain, equally likely, and impossible.” They
should be able to “predict the probability of
outcomes of simple experiments and test the
predictions.” They should “understand that the
measure of the likelihood of an event can be
represented by a number from 0 to 1.” (p. 400)
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Slide 9.5- 2
NCTM Standard: Data Analysis and
Probability
6–8: Children should “understand and use
appropriate terminology to describe
complementary and mutually exclusive
events.” They should be able “to make and
test conjectures about the results of
experiments and simulations.” They should be
able to “compute probabilities of compound
events using methods such as organized lists,
tree diagrams, and area models.” (p. 401)
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Slide 9.5- 3
9-5 Using Permutations and Combinations
in Probability
 Permutations of Unlike Objects
 Permutations Involving Like Objects
 Combinations
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Permutations of Unlike Objects
Permutation
An arrangement of things in a definite order
with no repetitions
Fundamental Counting Principle
If an event M can occur in m ways and, after M
has occurred, event N can occur in n ways,
then event M followed by event N can occur in
m · n ways.
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Definition
n
P
r
=
n factorial
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Permutations of Objects in a Set
n
P
r
=
In a set of n elements, the number of ways to
choose elements from the set in order, the
permutations of n objects taken r at a time, is given
by
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Slide 9.5- 7
Example 9-17
a. A baseball team has nine players. Find the
number of ways the manager can arrange the
batting order.
b. Find the number of ways of choosing three
initials from the alphabet if none of the letters
can be repeated.
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Permutations Involving Like Objects
If a set contains n elements, of which r1 are of one
kind, r2 are of another kind, and so on through rk,
then the number of different arrangements of all n
elements is equal to
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Example 9-18
Find the number of rearrangements of the letters in
each of the following words:
a. bubble
b. statistics
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Combinations
Combination
an arrangement of things in which the order
makes no difference
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Combinations
To find the number of combinations possible in a
counting problem, find the number of permutations
and then divide by the number of ways in which
each choice can be arranged.
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Slide 9.5- 12
Example 9-19
The Library of Science Book Club offers 3 free
books from a list of 42. If you circle 3 choices from
a list of 42 numbers representing the book on a
postcard, how many possible choices are there?
Order is not important, so this is a combination
problem.
There are 42 · 41 · 40 ways to choose the free books.
The three circled numbers can be arranged in
3 · 2 · 1 ways.
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Example 9-19
(continued)
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Example 9-20
At the beginning of the second quarter of a
mathematics class for elementary school teachers,
each of the class’s 25 students shook hands with
each of the other students exactly once. How
many handshakes took place?
Since the handshake between persons A and B is
the same as that between persons B and A, this is
a problem of choosing combinations of 25 people
2 at a time.
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Slide 9.5- 15
Example 9-21
Given a class of 12 girls and 10 boys, answer each
of the following:
a. In how many ways can a committee of 5
consisting of 3 girls and 2 boys be chosen?
The girls can be chosen in 12C3 ways.
The boys can be chosen in 10C2 ways.
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Example 9-21
(continued)
By the Fundamental Counting Principle, the total
number of committees is
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Example 9-21
(continued)
b. What is the probability that a committee of 5,
chosen at random from the class, consists of 3
girls and 2 boys?
The total number of committees of 5 is
22C5 = 26,334.
From part (a), we know that there are 9900
ways to choose 3 girls and 2 boys.
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Slide 9.5- 18
Example 9-21
(continued)
c. What is the probability that a committee of 5,
chosen at random from the class, consists of 3
girls and 2 boys?
The total number of ways to choose 5 girls and
0 boys from the 12 girls in the class is
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Slide 9.5- 19
Example 9-21
(continued)
d. What is the probability that a committee of 5,
chosen at random from the class, consists of
only girls?
The total number of committees of 5 is
22C5 = 26,334.
From part (c), we know that there are 792 ways
to choose 5 girls and 0 boys.
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