Mathematics 7
April 15, 2008
Template for Examination #1
1
. A box contains
N
tags numbered from
1
to
N
. In how many ways can
I select
k
of the tags?
2
. There are
b
boys and
a
girls in a group that forms a line. How many
boygirl patterns can be formed if at one end of the line there must be a boy
and at the other end there must be a girl?
3
. Evaluate
v
X
k
=
u
(
ak
+
b
)
4
.
Consider the data set
{
x
1
,
· · ·
, x
s
}
, where
x
1
=
·
,
x
2
=
·
,
· · ·
and
x
s
=
·
. Compute
(
i
)
the sample mean,
x
, and
(
ii
)
the sample median,
µ
.
(
iii
)
s
X
i
=1

x
i
−
µ

and
(
ii
)
s
X
j
=1
(
x
i
−
x
)
2
.
5
. Suppose a box contains
m
white balls and
n
black balls. Suppose I
select
w
balls at random. What is the probability that there are
z
white balls
in the sample?
6
. Suppose
m
X
i
=1
a
i
=
t
. Evaluate
m
X
i
=1
sa
i
.
7
. Evaluate
¡
N
k
¢
.
8
.
In a game in which there are
M
equally likely outcomes, suppose that
an event
E
can occur in
a
of the equally likely individual outcomes, and an
event
F
can occur in
b
equally likely individual outcomes. If these two events
are disjoint, compute the probability that at least one of these events occurs.
9
. Solve problem
8
when the two events have exactly
k
individual out
comes in common.
10
. If
A
and
B
are independent events, and if
P
(
A
) =
s
and
P
(
B
) =
t
,
compute the value of
P
(
A
∩
B
)
.
11
. If
A
and
B
are events, and if
P
(
A
∩
B
) =
c
and if
P
(
B
) =
d
, compute
P
(
A

B
)
.
12
. Suppose
P
(
D
) =
e
. Compute
P
(
D
c
)
.
1
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13
.
Suppose
A
and
B
are independent events, where
P
(
A
) =
u
and
P
(
B
) =
v
. Compute
P
(
A
∩
B
c
)
.
14
. Suppose a box contains
m
+
n
+
q
tags.
m
of the tags are numbered
a
,
n
of the tags are numbered
b
and
q
of the tags are numbered
c
. A tag is
drawn at random from the box. Let
X
denote the number on the tag drawn.
(
i
)
What is the range of
X
?
(
ii
)
Find
P
([
X
=
x
])
for all values of
x
in
range
(
X
/
)
.
15
. An urn contains three rags, numbered
a
,
b
and
c
. One selects
2
tags
at random from the three. Let
Z
denote the sum of the numbers on the two
tags selected. Find
P
([
Z
=
z
])
for all values of
z
in
range
(
Z
)
.
Mathematics 7
Template for Exam 2
May 6, 2008
No calculators
No scratch paper!
1. Let
X
be a random variable whose distribution is
Bin
(
n, p
)
.
(
i
)
What is the range of
X
?
(
ii
)
Find the density of
X
.
(
iii
)
Compute
P
([
X
≤
k
])
.
(
iv
)
Compute
P
([
a
≤
X
≤
b
])
. (Leave your answers as reduced fractions.)
2. If
Z
is a random variable whose distribution is
Bin
(
n, p
)
, and if
P
([
Z
≤
t
]) =
β
, compute
P
([
Z
≥
t
+ 1])
.
3. An urn contains
N
tags numbered from
1
to
N
. Let
Y
denote the sum
of the numbers of
n
of them selected at random.
(
i
)
What is the smallest number in
range
(
Y
)
?
(
ii
)
What is the largest number in
range
(
Y
)
?
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 Spring '08
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 Math, Statistics, equally likely outcomes

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