COMPANION FOR CHAPTER 1
OF
FUNDAMENTALS
OF PROBABILITY
WITH STOCHASTIC PROCESSES
FOURTH EDITION
SAEED GHAHRAMANI
Western New England University
Springfield, Massachusetts, USA
A CHAPMAN & HALL BOOK
,C ontents
1 Axioms of Probability 3
1A Additional Examples 3
1B Applications of Probability to Genetics 8
Answers to Odd-Numbered Exercises of Section 1B 13
, Section 1A Additional Examples 3
Chapter 1
A xioms of Probability
1A ADDITIONAL EXAMPLES
Example 1a In a mathematics department of 46 voting faculty members, there are two
candidates, Ryan and Nolan, running for the chair position. Assuming that each of the
46 faculty members votes for exactly one of the two candidates, and the candidate with a
majority wins, (a) define a sample space for the outcome of the election; (b) describe the
event that Nolan receives at least three times as many votes as Ryan.
Solution: (a) Let x be the number of faculty members who vote for Ryan. Then 46 − x
is the number of those who vote for Nolan. A sample space is
S = {(x, 46 − x) : x = 0, 1, 2, . . . 46}.
(b) The event that Nolan receives at least three times as many votes as Ryan is
E = (0, 46), (1, 45), (2, 44), . . .(11, 35) .
Example 1b Three numbers a, b, and c are selected randomly from the intervals (−1, 1),
(0, 1), (−1, 0), respectively. Define a sample space for this experiment and describe the
event that the quadratic equation ax 2 + bx + c = 0 has two distinct real roots.
Solution: Let R be the set of all real numbers. Then the sample space is
(a, b, c) ∈ R × R × R : − 1 < a < 1, 0 < b < 1, −1 < c < 0 ,
and the desired event is E = (a, b, c) ∈ S : b2 > 4ac , since the quadratic equation
ax2 + bx + c = 0 has two distinct real roots if and only if its discriminant, b 2 − 4ac, is
strictly positive.
Example 1c Define a sample space for the experiment of flipping a coin 12 times, and
describe the following events:
(a) the fifth flip is heads;
(b) at least one outcome is heads.
, Section 1A Additional Examples 4
Solution: A sample space for this experiment is
S = ω1 ω2 · · · ω12 : ωi = H or T, 1 ≤ i ≤ 12 .
The event that the fifth flip is heads is
A = ω1 ω2 ω3 ω4 Hω6 · · · ω12 : ωi = H or T, 1 ≤ i ≤ 12, i = 5 .
Let
A1 = Hω2 ω3 · · · ω12 : ωi = H or T, 2 ≤ i ≤ 12 ;
for 2 ≤ j ≤ 11, let
Aj = ω1 ω2 · · · ωj−1 Hωj+1 · · · ω12 : ωi = H or T, 1 ≤ i ≤ 12, i = j ;
and let
A12 = ω1 ω2 · · · ω11 H : ωi = H or T, 1 ≤ i ≤ 11 .
The event of at least one heads is 12
j=1 Aj .
Example 1d A farmer decides to build a pen in the shape of a triangle for his chickens.
He sends his son out to cut the lumber and the boy, without taking any thought as to the
ultimate purpose, makes two cuts at two points selected at random. (a) Describe a sample
space for this experiment. (b) Describe the event that the resulting three pieces of lumber
can be used to form a triangular pen.
Solution: (a) Let x be the distance from the left end of the lumber to the leftmost break.
Let y be the distance
from the left end
of the lumber to the rightmost break. The sample
space is S = (x, y) : 0 < x < y < .
(b) The three pieces x, y − x, and − y form a triangle if and only if
x < (y − x) + ( − y)
(y − x) < x + ( − y)
− y < x + (y − x),
or, equivalently,
x< , y <x+ , and y > .
2 2 2
So the event that the resulting three pieces of lumber can be used to form a triangular pen
is E = (x, y) : 0 < x < < y < x + .
2 2
Example 1e Travis picks up darts to shoot toward an 18 -diameter dartboard aiming at
the bullseye on the board. Consider a rectangular coordinate system with origin at the center
of the dartboard. The face of the dartboard is divided into 20 equal sectors (wedged-shaped
slices) with the positive x-axis being a straight side of the first sector (see Figure 1a). Define
a sample space for the point at which a dart hits the board. For this sample space, describe
the event that the dart hits a sector that has the positive y-axis as a side.
OF
FUNDAMENTALS
OF PROBABILITY
WITH STOCHASTIC PROCESSES
FOURTH EDITION
SAEED GHAHRAMANI
Western New England University
Springfield, Massachusetts, USA
A CHAPMAN & HALL BOOK
,C ontents
1 Axioms of Probability 3
1A Additional Examples 3
1B Applications of Probability to Genetics 8
Answers to Odd-Numbered Exercises of Section 1B 13
, Section 1A Additional Examples 3
Chapter 1
A xioms of Probability
1A ADDITIONAL EXAMPLES
Example 1a In a mathematics department of 46 voting faculty members, there are two
candidates, Ryan and Nolan, running for the chair position. Assuming that each of the
46 faculty members votes for exactly one of the two candidates, and the candidate with a
majority wins, (a) define a sample space for the outcome of the election; (b) describe the
event that Nolan receives at least three times as many votes as Ryan.
Solution: (a) Let x be the number of faculty members who vote for Ryan. Then 46 − x
is the number of those who vote for Nolan. A sample space is
S = {(x, 46 − x) : x = 0, 1, 2, . . . 46}.
(b) The event that Nolan receives at least three times as many votes as Ryan is
E = (0, 46), (1, 45), (2, 44), . . .(11, 35) .
Example 1b Three numbers a, b, and c are selected randomly from the intervals (−1, 1),
(0, 1), (−1, 0), respectively. Define a sample space for this experiment and describe the
event that the quadratic equation ax 2 + bx + c = 0 has two distinct real roots.
Solution: Let R be the set of all real numbers. Then the sample space is
(a, b, c) ∈ R × R × R : − 1 < a < 1, 0 < b < 1, −1 < c < 0 ,
and the desired event is E = (a, b, c) ∈ S : b2 > 4ac , since the quadratic equation
ax2 + bx + c = 0 has two distinct real roots if and only if its discriminant, b 2 − 4ac, is
strictly positive.
Example 1c Define a sample space for the experiment of flipping a coin 12 times, and
describe the following events:
(a) the fifth flip is heads;
(b) at least one outcome is heads.
, Section 1A Additional Examples 4
Solution: A sample space for this experiment is
S = ω1 ω2 · · · ω12 : ωi = H or T, 1 ≤ i ≤ 12 .
The event that the fifth flip is heads is
A = ω1 ω2 ω3 ω4 Hω6 · · · ω12 : ωi = H or T, 1 ≤ i ≤ 12, i = 5 .
Let
A1 = Hω2 ω3 · · · ω12 : ωi = H or T, 2 ≤ i ≤ 12 ;
for 2 ≤ j ≤ 11, let
Aj = ω1 ω2 · · · ωj−1 Hωj+1 · · · ω12 : ωi = H or T, 1 ≤ i ≤ 12, i = j ;
and let
A12 = ω1 ω2 · · · ω11 H : ωi = H or T, 1 ≤ i ≤ 11 .
The event of at least one heads is 12
j=1 Aj .
Example 1d A farmer decides to build a pen in the shape of a triangle for his chickens.
He sends his son out to cut the lumber and the boy, without taking any thought as to the
ultimate purpose, makes two cuts at two points selected at random. (a) Describe a sample
space for this experiment. (b) Describe the event that the resulting three pieces of lumber
can be used to form a triangular pen.
Solution: (a) Let x be the distance from the left end of the lumber to the leftmost break.
Let y be the distance
from the left end
of the lumber to the rightmost break. The sample
space is S = (x, y) : 0 < x < y < .
(b) The three pieces x, y − x, and − y form a triangle if and only if
x < (y − x) + ( − y)
(y − x) < x + ( − y)
− y < x + (y − x),
or, equivalently,
x< , y <x+ , and y > .
2 2 2
So the event that the resulting three pieces of lumber can be used to form a triangular pen
is E = (x, y) : 0 < x < < y < x + .
2 2
Example 1e Travis picks up darts to shoot toward an 18 -diameter dartboard aiming at
the bullseye on the board. Consider a rectangular coordinate system with origin at the center
of the dartboard. The face of the dartboard is divided into 20 equal sectors (wedged-shaped
slices) with the positive x-axis being a straight side of the first sector (see Figure 1a). Define
a sample space for the point at which a dart hits the board. For this sample space, describe
the event that the dart hits a sector that has the positive y-axis as a side.
