Section 7.4
195
12. Here
A
n
B
= 0,
so
P(A
n
B)
=
o. So
P(A
I
B)
=
0 and
P(B
I
A)
=
o.
A
and
B
are not
independent because
P(A)P(B)
=
;4
(~~)
while
P(A
n
B)
=
o.
A
and
B
are mutually exclusive
because
A
n
B
=
0.
13. [BB] Let
A
be the event "a 1 appears on the first die" and
B
the event "a 1 appears on the second die."
A
and
B
are not mutually exclusive because (1,1) belongs to
A
n
B.
However,
P(A)
=
~
=
P(B)
and
P(A
n
B)
=
3
1
6
=
P(A)P(B),
so
A
and
B
are independent.
14. A
and
B
are independent and mutually exclusive if and only if
An
B
=
0
and either
P(A)
=
0 or
P(B)
=
O. To see this, note first that if these conditions hold, then
A
and
B
are certainly mutually
exclusive and they are independent because
P(A
n
B)
=
0
=
P(A)P(B).
Conversely, if
A
and
B
are mutually exclusive, then
A
n
B
= 0,
so
P(A)P(B)
=
P(A
n
B)
=
0, implying
P(A)
=
0 or
P(B)
=
O.
15. (a) [BB] Here
P(A)
=
~,
P(B)
=
~,
P(A
n
B)
=
~.
So
P(A
n
B)
=I
P(A)P(B).
(b) [BB] Here
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 Summer '10
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 Exponential Function, Graph Theory, Mutually Exclusive, Initialisms, nd

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