This preview shows pages 1–2. Sign up to view the full content.
MATH 352 Assignment #2 Solutions
1. Use the Principle of Inclusion/Exclusion to solve each of the following problems:
(a) Find the probability that a permutation of
a
1
, a
1
, a
2
, a
2
, . . . , a
n
, a
n
will have no identical
symbols consecutive. The ﬁrst and last positions are not
considered to be consecutive.
Solution:
Let
A
i
be the event that the two
a
i
are consecutive. The desired probability
is the probability that none of the events occur, ie.
1

P
(
n
[
k
=1
A
i
) = 1

n
X
k
=1
(

1)
k

1
S
k
.
For any
i
, to calculate
P
(
A
i
), ‘glue’ together the two
a
i
, then permute the object
a
i
a
i
together with the other 2
n

2 objects: (2
n

1)!
/
(2!)
n

1
ways. Therefore
P
(
A
i
) =
(2
n

1)!
2
n

1
/
(2
n
)!
2
n
=
2
·
(2
n

1)!
(2
n
)!
.
This gives
S
1
=
n
·
2
·
(2
n

1)!
(2
n
)!
.
Using this same method, for any choice of
k
events,
P
(
A
i
1
A
i
2
. . . A
i
k
) =
(2
n

k
)!
2
n

k
/
(2
n
)!
2
n
=
2
k
·
(2
n

k
)!
(2
n
)!
,
and
S
k
=
±
n
k
²
·
2
k
·
(2
n

k
)!
(2
n
)!
.
The desired probability is 1

S
1
+
S
2
 ··· ±
S
n
.
(b) Suppose each of
n
sticks is broken into one long and one short part. The 2
n
parts are
then shuﬄed and rearranged into
n
pairs from which new sticks are formed. Find the
probabilitiy that
i. the parts will be joined into their original form.
Solution:
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '09
 M.Tao
 Probability

Click to edit the document details