SOLUTIONS to Homework Assignment 1
Probability Review
: Read Chapters 1 and 2 in the textbook,
Introduction to Proba
bility Models
, by Sheldon Ross. Please do the six problems marked with an asterisk and turn
them in next Tuesday. These problems are written out in case you do not yet have the textbook
(available in the Columbia Bookstore). Extra problems are provided in case you need extra
review. Solutions are provided for all these problems.
CHAPTER 1
*Problem 1.3.
A coin is to be tossed until a head appears twice in a row. What is the sample space for
this experiment? If the coin is fair, what is the probability that it will be tossed exactly four
times?
——————————————————————————
The sample space is a set with countably inﬁnitely many elements. The sample space is
the set
S
≡ {
(
e
1
,e
2
,...,e
n
) :
n
≥
2
}
,
where
e
i
∈ {
H,T
}
(i.e.,
e
i
is either heads or tails) and, in addition,
e
n
=
e
n

1
=
H
and,
for 1
≤
i
≤
n

2 (for
n
≥
3), if
e
i
=
H
, then necessarily
e
i
+1
=
T
. Here are initial ele
ments of
S
:
{
H,H
}
,
{
T,H,H
}
,
{
T,T,H,H
}
,
{
H,T,H,H
}
,
{
T,T,T,H,H
}
,
{
T,H,T,H,H
}
,
{
H,T,T,H,H
}
,
{
T,T,T,T,H,H
}
,...
——————————————————————————
*Problem 1.21
Suppose that 5% of men and 0
.
25% of women are colorblind. A colorblind person is
chosen at random. What is the probability of this person being male? (Assume that there are
an equal number of males and females.)
——————————————————————————
Let
C
be the event that a person chosen at random is colorblind. Let
M
be the event that
a random person is male. Let
M
c
be the event that a random person is female, the complement
of the event
M
. Then
P
(
M

C
) =
P
(
MC
)
P
(
C
)
=
P
(
C

M
)
P
(
M
)
P
(
C

M
)
P
(
M
) +
P
(
C

M
c
)
P
(
M
c
)
=
0
.
05
×
0
.
5
(0
.
05
×
0
.
5) + (0
.
0025
×
0
.
5)
=
20
21
.
——————————————————————————
Problem 1.22
———————————————————————–
The calculation becomes easy if we view it in a good way. One might observe that the
experiment has the form of tennis once deuce has been reached or table tennis (ping pong)