CSE21 FA11
Homework #7 Solutions
7.1.
A bin contains 4 red balls, 5 white balls and 6 blue balls. A random
subset S of 4 balls is removed (without replacement). Consider the following
3 events:
(1)
E
1
: S has exactly 2 red balls.;
(2)
E
2
: S has balls of all three colors;
(3)
E
3
: S has at least 2 blue balls.
Which of the pairs of events
E
1
, E
2
and
E
3
(if any) are independent? (Justify
your answer.)
Answer:
None of the pairs are independent. For example, take the pair
E
1
and
E
2
.
We have to compute
Pr
(
E
1
)
, P
(
E
2
) and
Pr
(
E
1
T
E
2
).
Since the
sample space has size
(
15
4
)
and all these choices are equally likely, then:
Pr
(
E
1
) =
(
4
2
)(
11
2
)
(
15
4
)
, Pr
(
E
2
) =
(
4
2
)(
5
1
)(
6
1
)
+
(
4
1
)(
5
2
)(
6
1
)
+
(
4
1
)(
5
1
)(
6
2
)
(
15
4
)
and
Pr
(
E
1
T
E
2
) =
(
4
2
)(
5
1
)(
6
1
)
(
15
4
)
.
Now we have to check if
Pr
(
E
1
T
E
2
) =
Pr
(
E
1
)
Pr
(
E
2
).
Since
this
doesn’t
hold, then the events
E
1
and
E
2
are
not
independent. Similar
calculations for the other two pairs show that they are also not independent.
7.2.
We are given an urn that has one red ball and two white balls in it. A
fair die is thrown. If the number shown is 1 or 2 then one red ball is added
to the urn. Otherwise three red balls are added to the urn. A ball is then
randomly drawn from the urn.
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 Fall '07
 Graham
 Probability theory, $10, $20, $40, $70

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