p. 2-11
Example (The Matching Problem).
Applications: (a) Taste Testing. (b) Gift Exchange.
Let
be all permutations
= (
i
1
, …,
i
n
) of 1, 2, …,
n
.
Thus, #
=
n
!.
Let
A
j
= {
:
i
j
=
j
}
and
,
Q
:
P
(
A
)=? (What would you expect when
n
is large?)
By symmetry,
Here,
σ
k
=
n
k
P
(
A
1
∩
···
∩
A
k
)
,
A
=
∪
n
i
=1
A
i
for
k
= 1, …,
n
.
p. 2-12
Proportion: If
A
1
,
A
2
, …, is a
partition
of
, i.e.,
1.
2.
A
1
,
A
2
, …, are mutually exclusive,
then, for any event
A
⊂
,
P
(
A
)=
∞
i
=1
P
(
A
∩
A
i
)
.
∪
∞
i
=1
A
i
=
Ω
,
So,
Note
: approximation accurate to 3 decimal places if
n
6.
P
(
A
σ
1
−
σ
2
+
+(
−
1)
n
+1
σ
n
=
n
k
=1
(
−
1)
k
+1
1
k
!
,
P
(
A
)=1
−
n
k
=0
(
−
1)
k
1
k
!
≈
1
−
1
e
⇒
P
(
A
c
)
≈
e
−
1