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Chapter 2 Probability
2.1 Sample Space
2.2 Events
2.3 Counting Sample Points
2.4 Probability of an Events
2.5 Additive Rules
2.6 Conditional Probability
2.7 Multiplication Rules
2.8 Bayes’s Rule
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2.1 Sample Space
For any random phenomenon in which there is
uncertainty concerning which of two or more possible
outcomes will result, we need to
1. Identify or define the process that generates the
outcomes
2. List all possible outcomes
3. Find or define probability of each outcome.
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Statistical Experiment: A process that generates
a data set.
Example 1: Rolling a die and observing the number
facing up.
There six possible outcomes: 1, 2, …, 6.
Example 2: Counting the number of the students in the
university who use an internet at 7:00 pm on
Monday.
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Sample Space: The set of all possible outcomes of
a statistical experiment, often denoted by the
symbol S.
In Example 1,
In Example 2,
S = { 1, 2, …, 6}
S = {0, 1, 2, …, N}
Sample point:
Each outcome in a sample space is called a
sample point.
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List all possible outcomes
Tree diagram: Sometimes, we can list all
possible outcomes with a tree diagram.
Example 2.2, page 23. An experiment consists
of flipping a coin and then flipping it second
time if a head occurs. If a tail occurs on the first
flip, then a die is tossed once. Figure 2.1, page
24.
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First outcome
Sample
point
T1
T2
T
T3
T4
T5
T6
HH
H
HT
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Example 2.3, page 24. During a space shot, the
primary computer system is backed up by two
secondary systems in succession . They operate
independently of one another.
Find the sample space of the readiness of three
systems at the time of launch.
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Primary
1st backup
2nd back up
y
yyy
y
y
n
n
yyn
y
yny
n
ynn
y
nyy
n
y
nyn
nny
n
nnn
y
n
n
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Describe the sample points by statements or by rules
(for a large or infinite number of sample points)
Example: The possible outcomes of an experiment are
the set of the cities in the world with a population over 1
million.
S = {x | x is a city with a population over 1 million}
Example: S is all points (x, y) on the boundary and
interior of a circle of radius 2 with center at the origin.
S = {(x, y) | x2  y 2  4
}
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2.2 Events
Often for an experiment, we are interested in the
occurrence of a certain event rather than in a single
sample point.
For example, rolling a die, we may be interested in
whether “the number is at least 3”.
A = {3, 4, 5, 6}.
And in the example of space shot, we are interested in
the event that “at least two systems are operable at the
time of launch”.
B = { yyy, yyn, yny, nyy}
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Event: is a subset of a sample space.
Example 2.4: Given the sample space
S = {t | t  0}
where t is the life time in years of a certain
electronic component, then the event that the
component fails before the end of the fifth year is
A = {t | 0  t < 5}. A⊂ S
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The relationship and operation of events
• Complement event: The complement event
of an event A with respect to S is the subset
of all elements of S that are not in A. We
denote it by A’.
A’ occurs if and only if A does not occur.
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Intersection of two events: The intersection of two
events A and B, denoted by AB, is the event containing
all elements that are common to A and B.
AB occurs if and only if both A and B occur.
Mutually exclusive, or Disjoint: If AB = , then we
say that A and B are disjoint or mutually exclusive.
A and A’ are always disjoint or mutually exclusive.
Events A1, A2, …, An are mutually exclusive if and only if
AiAj = (i ≠j).
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A and B are
disjoint
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Union: The union of the two events A and B, denoted
by AB, is the event containing all the elements that
belong to A or B or both.
AB occurs if either A or B occur or both occur.
Example: Consider a space shot in which a primary computer
system is backed up with two secondary computer systems.
S = {yyy, yyn, yny, ynn, nyy, nyn, nny, nnn}
A = {yyy, yyn, yny, ynn}
(primary system is operable)
B = {yyy, yyn, nyy, nyn}
(first backup is operable)
C = {yyy, yny, nyy, nny}
(second backup is operable)
Find A  B, A  B, (A  B)  C’
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Complement
Four commonly used properties of events
1. A  B = B  A
AB=BA
2. A (B  C) = (A  B)  C A(BC)=(AB)C
3. A (BC)=(A  B)(A  C)
A(B  C)=(AB) (AC)
4. (A  B)’=A’ B’
(AB)’=A’  B’
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