CEE 3770  Summer 2009
Homework 1 (due on 05/21)
1. (6 points) Let
E, F, G
be three events. Suppose that
U
=
E
∪
F
∪
G
. Find expressions for
the events that of
E, F, G
(a) only F occurs.
Solution:
E
∩
F
∩
G
(=
E
∩
G, sinceU
=
E
∪
F
∪
G
)
(b) both E and F occur, but not G.
Solution:
E
∩
F
∩
G
(c) at least one event occurs.
Solution:
E
∪
F
∪
G
(=
U
)
(d) at least two events occur.
Solution:
(
E
∩
F
)
∪
(
E
∩
G
)
∪
(
F
∩
G
)
(e) all three events occur.
Solution:
E
∩
F
∩
G
(f) none occur.
Solution:
E
∪
F
∪
G
(=
∅
), since ”none” is the opposite of at least one”
2. (6 points) A box contains 3 marbles (two red and one blue). Consider an experiment that
consists of taking one marble from the box, then replacing it in the box, and then drawing a
second marble from the box.
(a) What is the sample space?
Solution:
S
=
{
RR, RB, BR, BB
}
(b) Is it SSS(Simple Sample Space)? If not, construct a simple sample space.
Solution:
No. Clearly
BB
occurs with smaller probability (=1/9). A simple sample
space,
S
=
{
R
1
R
1
, R
1
R
2
, R
1
B, R
2
R
1
, R
2
R
2
, R
2
B, BR
1
, BR
2
, BB
}
(c) Find the probability that at least one blue marble is chosen.
Solution:
From
S
,
5
9
.
3. (6 points) Repeat the above when the second marble is drawn without replacing the first
marble.
(a) What is the sample space?
Solution:
S
=
{
RB, RR, BR
}
(b) Is it SSS(Simple Sample Space)? If not, construct a simple sample space.
Solution:
Yes.
Pr
(
RB
)
=
2
3
1
2
=
1
3
Pr
(
RR
)
=
2
3
1
2
=
1
3
Pr
(
BR
)
=
1
3
·
1 =
1
3
1
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(c) Find the probability that at least one blue marble is chosen.
Solution:
Since
S
is SSS, 2
/
3
4. (6 points) On a surprise quiz with two questions, the first question is multiple choice with
a,b,c and d (4 alternatives) while the second question is a true/false question. A student is
completely unprepared for the quiz and answers at random.
(a) Give a sample space for this experiment.
Solution:
S
=
{
aT, aF, bT, bF, cT, cF, dT, dF
}
(b) The instructor is interested in the number of points on this quiz.
Correct answer on
the first question carries 5 points and incorrect 1 point. Correct answer on the second
question carries 5 points, incorrect 3.
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 Spring '08
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 Probability theory, Column, How Many Ways

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