Download Section 2: Sample Space and Events

Probability and Statistics
Sample Space and Events
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Dr. Saeid Moloudzadeh
www.soran.edu.iq
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Contents
Descriptive Statistics
Axioms of Probability
Combinatorial Methods
Conditional Probability and
Independence
Distribution Functions and
Discrete Random Variables
Special Discrete Distributions
Continuous Random Variables
Special Continuous Distributions
Bivariate Distributions
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Probability and Statistics
•
•
•
•
Contents
Descriptive Statistics
Axioms of Probability
Combinatorial Methods
Conditional Probability and Independence
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Distribution Functions and Discrete Random Variables
•
•
•
•
Special Discrete Distributions
Continuous Random Variables
Special Continuous Distributions
Bivariate Distributions
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Chapter 1: Axioms of Probability
Context
• Sample Space and Events
• Axioms of Probability
• Basic Theorems
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Chapter 1: Axioms of Probability
Context
• Sample Space and Events
• Axioms of Probability
• Basic Theorems
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Section 2: Sample Space and Events
Definition (Experiment): An experiment is an act or
process of observation that leads to a single
outcome that cannot be predicted with certainty.
A sample point is the most basic outcome of an
experiment.
Definition (Sample space): The set of all possible
outcomes to a random experiment (all the
sample points) is called the sample space and is
denoted by S.
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Section 2: Sample Space and Events
Example 2.1: Some examples of experiments and their sample
spaces are as follows:
(a) If the outcome of the experiment is the gender of a child, then
S = {g, b}
where outcome g means that the child is a girl and b that it is
a boy.
(b) If the experiment consists of flipping two coins and noting
whether they land heads or tails, then :
S = {(H,H), (H, T), (T,H), (T, T)}
The outcome is (H, H) if both coins land heads, (H, T) if the first
coin lands heads and the second tails, (T, H) if the first is tails
and the second is heads, and (T, T) if both coins land tails.
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Section 2: Sample Space and Events
(c) If the experiment consists of tossing two dice, then the
sample space consists of the 36 points
S = {(i, j): i, j = 1, 2, 3, 4, 5, 6}
where the outcome (i, j) is said to occur if i appears on the
leftmost die and j on the other die.
Definition (Sample events): Any set of outcomes of the
experiment is called an event. That is, an event is a
subset of the sample space. Events will be denoted by the
capital letters A, B, C and so on.
We say that the event A occurs whenever the outcome is
contained in A.
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Section 2: Sample Space and Events
Example 2.2: In Example 2.1 (a), if A = {g}, then A is
the event that the child is a girl. Similarly,
if B = {b}, then B is the event that the child is a boy.
In Example 2.1 (b), if A = {(H, H), (H, T)}, then A is
the event that the first coin lands on heads.
In Example 2.1 (c), if
A = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}
then A is the event that the sum of the dice is 7.
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Section 2: Sample Space and Events
In the study of probability theory the relations between
different events of an experiment play a central role. In
the remainder of this section we study these relations.
In all of the following definitions the events belong to a
fixed sample space S.
Definition:
• Subset: An event A is said to be a subset of the event B
if, whenever A occurs, B also occurs. This means that all
of the sample points of A are contained in B. Hence
considering A and B solely as two sets, A is a subset of
B in the usual set-theoretic sense: that is, A  B .
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Section 2: Sample Space and Events
Equality: Events A and B are said to be equal if the
occurrence of A implies the occurrence of B, and
vice versa; that is, if A  B and B  A , hence A=B.
Intersection: An event is called the intersection of
two events A and B if it occurs only whenever A
and B occur simultaneously. In the language of
sets this event is denoted by AB or A B because
it is the set containing exactly the common points
of A and B.
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Section 2: Sample Space and Events
• Union: An event is called the union of two events A
and B if it occurs whenever at least one of them
occurs. This event is A B since all of its points are
in A or B or both.
• Complement: An event is called the complement of
the event A if it only occurs whenever A does not
c
A
occur. The complement of A is denoted by .
• Difference: An event is called the difference of two
events A and B if it occurs whenever A occurs but B
does not. The difference of the events A and B is
denoted by A-B.
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Section 2: Sample Space and Events
• Certainty: An event is called certain if its occurrence is
inevitable. Thus the sample space is a certain event.
• Impossibility: An event is called impossible if there is
certainty in its nonoccurrence.
Therefore, the empty set  , is an impossible event.
• Mutually Exclusiveness : If the joint occurrence of two
events A and B is impossible, we say that A and B are
mutually exclusive or disjoint. Thus A and B are mutually
exclusive if the occurrence of A precludes the occurrence of
B, and vice versa. Since the event representing the joint
occurrence of A and B is AB, their intersection, A and B, are
mutually exclusive if AB =  .
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Section 2: Sample Space and Events
Shaded region in the following figure presents Venn
diagrams for AB, A B , A and B are disjoint events.
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Section 2: Sample Space and Events
Example 2.3: At a busy international airport,
arriving planes land on a first-come, first-served
basis. Let
E = there are at least five planes waiting to land,
F = there are at most three planes waiting to land,
H = there are exactly two planes waiting to land.
Then
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Section 2: Sample Space and Events
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Section 2: Sample Space and Events
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Section 2: Sample Space and Events
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