Solutions to Homework 7, Math 55
1. The probability that Beatrix starts with the crown and throws it to Andrew is
1
3
·
1
2
=
1
6
; similarly, the
probability that Charles starts with the crown and throws it to Andrew is
1
6
. Therefore, the probability that
Andrew has the crown after one turn is
1
6
+
1
6
=
1
3
. Similarly, the probability that Charles has the crown
after one turn is
1
3
from Andrew, and
1
3
·
1
2
=
1
6
from Beatrix, so the total probability is
1
3
+
1
6
=
1
2
. On the
other hand, if Beatrix gets the crown, it must be from Charles; therefore, the probability that Beatrix has
the crown after one turn is
1
3
·
1
2
=
1
6
.
2. We have
p
(
A
) =
4
52
=
1
13
;
p
(
B
) =
13
52
=
1
4
; and
p
(
C
) =
26
52
=
1
2
.
Now
p
(
A
∩
B
) =
1
52
=
p
(
A
)
p
(
B
), and
p
(
A
∩
C
) =
2
52
=
p
(
A
)
p
(
C
), but
p
(
B
∩
C
) =
13
52
=
p
(
B
)
p
(
C
). Thus,
A
and
B
are independent, and so are
A
and
C
, but
B
and
C
are not independent.
If we add a joker to the deck, then
p
(
A
) =
4
53
;
p
(
B
) =
13
53
; and
p
(
C
) =
26
53
. However,
p
(
A
∩
B
) =
1
53
=
p
(
A
)
p
(
B
);
p
(
A
∩
C
) =
2
53
=
p
(
A
)
p
(
C
); and
p
(
B
∩
C
) =
13
53
=
p
(
B
)
p
(
C
). Thus, in this case, no pair of events
from
A, B, C
is independent.
3. We have
p
(
A
∩
B
) =
p
(
S

(
A
∪
B
)) = 1

p
(
A
∪
B
) = 1

(
p
(
A
) +
p
(
B
)

p
(
A
∩
B
)) = 1

p
(
A
)

p
(
B
) +
p
(
A
)
p
(
B
) = (1

p
(
A
))(1

p
(
B
)) =
p
(
A
)
p
(
B
). Therefore,
A
and
B
are independent.
Similarly,
p
(
A
∩
B
) =
p
(
A

(
A
∩
B
)) =
p
(
A
)

p
(
A
∩
B
) =
p
(
A
)

p
(
A
)
p
(
B
) =
p
(
A
)(1

p
(
B
)) =
p
(
A
)
p
(
B
),
so
A
and
B
are independent. (Note that in the step
p
(
A

(
A
∩
B
)) =
p
(
A
)

p
(
A
∩
B
), we need to use
the fact that
A
∩
B
⊆
A
.) However, if
A
and
A
are independent, then
p
(
A
)
p
(
A
) =
p
(
A
∩
A
) =
p
(
∅
) = 0,
which implies
p
(
A
) = 0 or
p
(
A
) = 0, and in the latter case,
p
(
A
) = 1

p
(
A
) = 1. Thus,
A
and
A
are not
independent unless
p
(
A
) = 0 or
p
(
A
) = 1.
4. Let
E
be the event that the number is divisible by 3, and
F
be the event that the number is divisible by 5.
Then
p
(
E
) =
300
900
=
1
3
; and
p
(
F
) =
180
900
=
1
5
. Also,
E
∩
F
is the event that the number is divisible by 15,
which has probability
60
900
=
1
15
=
p
(
E
)
p
(
F
). Therefore, the two given events are independent.
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 Spring '08
 STRAIN
 Math, Probability, Variance, Probability theory, probability density function

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