(c) We want to find
m
so that
F
(
m
) = 1
/
2. Looking at the CDF above, we see
F
(1
/
2) = 1
/
4,
while
F
(1) = 1, so 1
/
2
< m <
1. Thus we must solve (3
m
−
1)
/
2 = 1
/
2. This gives
m
= 2
/
3.
2. (2+4+4 pts) You have a bag containing three marbles numbered 0, 1, and 2. You sample 5
times from the bag, with replacement.
(a) How many outcomes are in my sample space for this experiment?
(b) What is the probability that you never pick the same marble twice in a row?
(c) What is the probability that the numbers that you picked add up to 5?
Solution.
(a) 3
5
.
(b) 3
×
2
4
/
3
5
= (2
/
3)
4
. There are 3 options for the first pick and then, to avoid repeating
the same pick twice in a row, only 2 options for the second pick, and likewise for the 3rd,
4th, and 5th picks.
(c) There are three cases to consider: (I) 2 + 2 + 1 + 0 + 0, (II) 2 + 1 + 1 + 1 + 0, or (III)
1+1+1+1+1. For case (I), there are 5
×
(
4
2
)
possible orderings for these picks: five options
for where to pick the '1', then four-choose-two options for where to pick the two '2's. For
case (II), there are 5
×
(
4
3
)
orderings, and for case (III) there is only one ordering. Thus, the
total probability is
5
×
4
2
+ 5
×
4
3
+ 1
/
3
5
.
3. (3+3+4 pts) We roll a 4-sided die 5 times. Let
X
denote the number of '1's rolled and let
Y
denote the number of '2's.
(a) Compute
P
{
X
= 3
}
.
(b) Compute
P
(
{
X
= 0
} ∪ {
Y
= 0
}
)
.
(c) Are
X
and
Y
are independent random variables? Prove or disprove your answer.
Solution.
(a)
X
∼
Binomial(5
,
1
/
4), so
P
{
X
= 3
}
=
(
5
3
)(
1
4
)
3
(
3
4
)
2
.
(b)
P
{
X
= 0
}
=
P
{
Y
= 0
}
=
(
3
4
)
5
, and
P
(
{
X
= 0
} ∩ {
Y
= 0
}
) =
(
2
4
)
5
, since
X
and
Y
both
being 0 would mean all die rolls are '3's and '4's. Thus, by inclusion-exclusion,
Pr
(
{
X
= 0
} ∪ {
Y
= 0
}
)
=
3
4
5
+
3
4
5
−
2
4
5
.
(c)
P
{
X
= 5
}
and
P
{
Y
= 5
}
are both non-zero (it is possible to roll all '1's or all '2's),
but
P
(
{
X
= 5
} ∩ {
Y
= 5
}
) = 0, as it is impossible to get five '1's
and
five '2's in just five
die rolls. Thus,
P
(
{
X
= 5
} ∩ {
Y
= 5
}
)
̸
=
P
{
X
= 5
}
P
{
Y
= 5
}
, so these variables are not
independent.