p. 6-11
n
n
1
,
···
,n
m
=
n
!
n
1
!
×···×
n
m
!
ways
.
Example: MISSISSIPPI
11
4
,
1
,
2
,
4
=
11!
4!1!2!4!
.
Example (Die Rolling).
Q
: If a balanced (6-sided) die is rolled 12 times,
P
(each face appears twice)=??
Sample space of rolling the die once (basic experiment):
0
= {1, 2, 3, 4, 5, 6}.
The sample space for the 12 trials is
=
0
×
×
0
=
0
12
An outcome
∈
is
=(
i
1
,
i
2
, …,
i
12
), where
1
≤
i
1
, …,
i
12
≤
6.
There are 6
12
possible outcomes in
, i.e., #
= 6
12
.
Among all possible outcomes, there are
of which each face appears twice.
P
(each face appears twice) =
12
2
,
2
,
2
,
2
,
2
,
2
=
12!
(2!)
6
12!
(2!)
6
1
6
12
.
p. 6-12
Generalization.
Consider a basic experiment which can result in one of
m
types of outcomes. Denote its sample space as
0
= {1, 2, …,
m
}.
Let
p
i
=
P
(outcome
i
appears),
then,
(i)
p
1
, …,
p
m
≥
0, and
(ii)
p
1
+
+
p
m
= 1.
Repeat the basic experiment
n
times. Then, the sample
space for the
n
trials is
=
0
×
×
0
=
0
n
Let
X
i
= # of trials with outcome
i
,
i
=1, …,
m
,
Then,
(i)
X
1
, …,
X
m
:
→
R
, and
(ii)
X
1
+
+
X
m
=
n
.
The joint pmf of
X
1
, …,
X
m
is
p
X
(
x
1
,...,x
m
)=
P
(
X
1
=
x
1
,...,X
m
=
x
m
)
=
n
x
1
,
···
,x
m
p
x
1
1
×···×
p
x
m
m
.
for
x
1
, …,
x
m
≥
0 and
x
1
+
+
x
m
=
n
.