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ISyE 3770 — Test 1
a
Solutions — Fall 2010
(revised 10/1/10)
This test is 55 minutes long. You are allowed one cheat sheet.
1. TRUE or FALSE?
If
A
1
,A
2
,...,A
n
are disjoint events, then
P
(
∪
n
i
=1
A
i
) =
∑
n
i
=1
P
(
A
i
).
Solution:
TRUE.
P
(
A
i
∩
A
j
) = 0 for all
i,j
.
♦
2. Suppose that
P
(It snows today

It’s cold outside) = 0
.
8
and
P
(It snows today and it’s cold outside) = 0
.
5
.
Find the probability that it’s cold outside.
Solution:
P
(Snow

Cold) =
P
(Snow and Cold)
P
(Cold)
P
(Cold) =
P
(Snow and Cold)
P
(Snow

Cold)
=
5
8
♦
3. If
P
(
A
) = 0
.
3,
P
(
B
) = 0
.
5, and
P
(
C
) = 0
.
5, and
A
,
B
, and
C
are independent,
ﬁnd the probability that
all three
of
A
,
B
, and
C
occur.
Solution:
P
(
A
∩
B
∩
C
) =
P
(
A
)
P
(
B
)
P
(
C
) = 0
.
075.
♦
4. Consider a box of 10 sox — 6 blue and 4 red. Pick two sox without replacement.
What’s the probability that the second sock is red?
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Solution:
P
(2nd is Red) =
P
(1st is Blue, 2nd is Red) +
P
(1st is Red, 2nd is Red)
=
±
6
10
¶±
4
9
¶
+
±
4
10
¶±
3
9
¶
=
2
5
♦
5. TRUE or FALSE? If
A
and
B
are independent, then
A
and
B
are disjoint.
Solution:
FALSE. For example, if
A
and
B
are independent,
P
(
A
) = 0
.
2
,
P
(
B
) = 0
.
5 and
P
(
A
∩
B
) = (0
.
2)(0
.
5) = 0
.
1
6
= 0.
♦
6. What is 0! (zero factorial)?
Solution:
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 Spring '07
 goldsman

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