196
Solutions
to
Exercises
20. [BB] We
use
induction
on
n,
the strong form.
For
n
=
1, the result is clear.
If
n
=
2,
P(A
I
U
A2)
=
P(A
I)
+
P(A
2)

P(A
I
n
A2)
~
P(A
I)
+
P(A2)'
Now assume
n
>
2
and the result holds for all
integers
k
with
1
~
k
<
n.
We have
P(A
I
U
A2
U
...
U
An)
=
P((A
I
U
A2
U·
U
AnI)
U
An)
~
P(A
I
U
A2
U
...
U
AnI)
+
P(An)
~
(P(A
I)
+
P(A
2)
+ ... +
P(A
n I)
+
P(An)
By
the Principle
of
Mathematical Induction, the result follows.
using the
k
=
2 case
using the
k
=
n

1
case.
21. We use induction
on
n,
the strong form.
For
n
=
1, the result is clear.
If
n
=
2,
P(A
I
n
A2)
=
P(A
I)
+
P(A
2)

P(A
I
U
A2)
~
P(A
I)
+
P(A
2)

1.
Now
assume
n
>
2
and
the result holds for
all integers
k
with
1
~
k
<
n.
We have
P(A
I
n
A2
n ...
nAn)
=
P((A
I
n
A2
n·
n
AnI}
nAn)
~
P(A
I
n
A2
n ...
nAnI)
+
P(An)

1
~
(P(A
I)
+
P(A
2)
+ ... +
P(A
n I)

(n

2)
+
P(An)

1
=
P(A
I)
+
P(A
2)
+ ... +
P(An)

(n

1).
By
the Principle
of
Mathematical Induction, the result follows.
22.
(a)
365(364)(363)···
(365 
(n

1))
(365)n
using the
k
=
2 case
using the
k
=
n

1
case
(b)
When
n
=
23, the fraction in (a) is .492703 and this value is closer to .5 than with any other
n.
23.
nPx
is the probability
of
the event
"a
person aged
x
survives
to
age
x
+
n"
while
mPx+n
is the
probability
of
the event
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 Summer '10
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 Graph Theory, Integers, Trigraph, strong form, athematical Induction

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