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T. Liggett
Mathematics 170A – Midterm 1 Solutions
October 18, 2010
(20) 1. Suppose
A
and
B
are events with
P
(
A
) =
3
4
and
P
(
B
) =
1
3
.
(a) What is the largest possible value for
P
(
A
∩
B
)? Explain.
Solution:
P
(
A
∩
B
)
≤
P
(
B
) =
1
3
, and this value is attained if
B
⊂
A
.
(b) What is the smallest possible value for
P
(
A
∩
B
)? Explain.
Solution:
P
(
A
∩
B
) =
P
(
A
) +
P
(
B
)
−
P
(
A
∪
B
)
≥
3
4
+
1
3
−
1 =
1
12
, and this
value is attained if, for example, Ω =
{
1
,
2
, ...,
12
}
with each point having
weight
1
12
,
A
=
{
1
, ...,
9
}
, B
=
{
9
, ...,
12
}
.
(10) 2. Three cups, colored red, white and blue respectively, are placed
on three saucers, also colored red, white and blue, at random. Find the
probability that no cup is on the saucer of the same color.
Solution:
There are 3! = 6 possible arrangements, and two have no color
matches, so the answer is
1
3
.
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