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Stat 311 Introduction to Mathematical
Statistics
Zhengjun Zhang
Department of Statistics, University of Wisconsin, Madison, WI 53706, USA
September 2224, 2009
Example
•
Suppose we have 6 cards
{
A,K,Q,J,10,9
}
and we select a pair of diﬀerent cards from the
list without replacement.
•
Q:
P
(
A
) =
P
(
{
both letter cards selected
}
)
•
Solve: List all possible pairs
Problems
•
If the sample space has many items, this kind of work becomes tedious, we will use
counting method.
Combinatorial principles
•
The
science
of counting is called
combinatorics
.
•
Many problems in probability theory require that we count the number of ways that a
particular event can occur.
•
For this, we study the topics of
permutations
and
combinations.
Counting rules
•
Multiplication rule 1: total number of outcomes of a sequence when each has
k
possi
bilities =
k
n
.
•
Multiplication rule 2: total number of outcomes of a sequence when each has a diﬀerent
number of possibilities
k
1
·
k
2
·
k
3
···
k
n
.
0
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In set notation
S
=
{
(
s
1
,...,s
n
) :
s
i
∈
S
i
}
=
S
1
× ··· ×
S
k

S

=

S
1
 × ··· × 
S
k

.
Example Suppose we ﬂip three coins and roll two fair sixsided dice. What is the probability
that all three coins come up heads, and both dice come up 6?
Example: Traveling salesman problem Start from city C0, will visit
n
other cities one by one
Counting formula 1 A
permutation
is an ordered arrangement of distinct items. The total
number of permutations of
n
distinct items is
n
(
n

1)
···
(2)(1) =
n
!
Theorem 3.1 The total number of permutations of a set
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