Transcript Lecture 2.3 - Sybil Nelson

2
Probability
Copyright © Cengage Learning. All rights reserved.
2.3
Counting Techniques
Copyright © Cengage Learning. All rights reserved.
Counting Techniques
When the various outcomes of an experiment are equally
likely (the same probability is assigned to each simple
event), the task of computing probabilities reduces to
counting.
Letting N denote the number of outcomes in a sample
space and N(A) represent the number of outcomes
contained in an event A,
(2.1)
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The Product Rule for Ordered Pairs
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The Product Rule for Ordered Pairs
Proposition
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Example 2.18
A family has just moved to a new city and requires the
services of both an obstetrician and a pediatrician. There
are two easily accessible medical clinics, each having two
obstetricians and three pediatricians.
The family will obtain maximum health insurance benefits
by joining a clinic and selecting both doctors from that
clinic. In how many ways can this be done?
Denote the obstetricians by O1, O2, O3, and O4 and the
pediatricians by P1, . . . ., P6.
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Example 2.18
cont’d
Then we wish the number of pairs (Oi, Pj) for which
Oi and Pj are associated with the same clinic.
Because there are four obstetricians, n1 = 4, and for each
there are three choices of pediatrician, so n2 = 3.
Applying the product rule gives N = n1n2 = 12 possible
choices.
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The Product Rule for Ordered Pairs
In many counting and probability problems, a configuration
called a tree diagram can be used to represent pictorially
all the possibilities.
The tree diagram associated with Example 2.18 appears in
Figure 2.7.
Tree diagram for Example 18
Figure 2.7
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A More General Product Rule
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A More General Product Rule
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Example: License Plates
Suppose the state of South Carolina creates 6 digit license
plates with three letters and then three numbers. How
many license plates can it make?
How many license plates can it make if they do not want to
repeat numbers or letters?
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Permutations and Combinations
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Permutations and Combinations
Consider a group of n distinct individuals or objects
(“distinct” means that there is some characteristic that
differentiates any particular individual or object from any
other).
How many ways are there to select a subset of size k from
the group?
For example, if a Little League team has 15 players on its
roster, how many ways are there to select 9 players to form
a starting lineup?
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Permutations and Combinations
Or if a university bookstore sells ten different laptop
computers but has room to display only three of them, in
how many ways can the three be chosen?
An answer to the general question just posed requires that
we distinguish between two cases. In some situations, such
as the baseball scenario, the order of selection is
important.
For example, Angela being the pitcher and Ben the catcher
gives a different lineup from the one in which Angela is
catcher and Ben is pitcher.
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Permutations and Combinations
Often, though, order is not important and one is interested
only in which individuals or objects are selected, as would
be the case in the laptop display scenario.
Definition
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Permutations and Combinations
Proposition
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Permutations and Combinations
Proposition
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Example
A local college is investigating ways to improve the
scheduling of student activities. A fifteen-person committee
consisting of five administrators, five faculty members, and
five students is being formed. A five-person subcommittee
is to be formed from this larger committee. The chair and
co-chair of the subcommittee must be administrators, and
the remainder will consist of faculty and students. How
many different subcommittees could be formed?
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Example
Two sub-choices:
1. Choose two administrators.
2. Choose three faculty and students.
Number of choices:
10 
5 P2  
  20 120  2400
3
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Arrangments
How many “words” can be formed from the word
MISSISSIPPI?
By words I just mean arrangements of letters…
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Example 2.22
A particular iPod playlist contains 100 songs, 10 of which
are by the Beatles.
Suppose the shuffle feature is used to play the songs in
random order (the randomness of the shuffling process is
investigated in “Does Your iPod Really Play Favorites?”
What is the probability that the first Beatles song heard is
the fifth song played?
In order for this event to occur, it must be the case that the
first four songs played are not Beatles’ songs (NBs) and
that the fifth song is by the Beatles (B).
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Example 2.22
cont’d
The number of ways to select the first five songs is
100(99)(98)(97)(96).
The number of ways to select these five songs so that the
first four are NBs and the next is a B is 90(89)(88)(87)(10).
The random shuffle assumption implies that any particular
set of 5 songs from amongst the 100 has the same chance
of being selected as the first five played as does any other
set of five songs; each outcome is equally likely.
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Example 2.22
cont’d
Therefore the desired probability is the ratio of the number
of outcomes for which the event of interest occurs to the
number of possible outcomes:
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