Probability
Definitions
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Experiment
A process by which an outcome is obtained.
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Sample space.
The set of all possible outcomes of an
experiment.
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Event
Any subset of the sample space.
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Sample Point
Any element of the sample space, an
outcome.
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Theoretical Probability
A probability of an event is the long term
relative frequency of the event.
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The Cardinal Number of a Set
The cardinal number of a set is the number
of elements in the set. The cardinal number
of the sample space is denoted by n(S).
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Example
A fair die is rolled. Find:
a) the sample space.
b) the event the number face up is even.
c) the probability of an even number is café up.
d) the probability of an odd number is café up.
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The Probability of an Event
The occurrence of an event:
We say an event E occurred, if any of its elements
occurred.
The number of ways E occur is equal to its cardinal
number.
The probability of an event E, written p(E), is given by
Where n(E) and n(S) are the cardinal numbers of the
event E and the sample space S respectively.
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Examples
A fair die is rolled. Find the following probabilities.
a)
P(2 face up)
b) P(number face up is even)
c)
P(x=2 or 3)
d) P(x=1,2,3,4,5,or 6)
e)
P(the number face up is odd or even)
f)
P(the number face up is odd and even)
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Set Operations
Definitions
1.
Union
The union of sets A and B, written
, is the
set whose elements are members of set A, or set
B, or both sets A and B.
2.
Intersection
The intersection of sets A and B, written
, is
the set whose elements are members of Set A and
set B.
3.
Complement
The complement of a set A, written ,
is the set whose element are members of the
sample space S, but they are not elements of the
set A.
4.
The Empty set
The empty set is a set that contains no elements,
and it is denoted by the Greek letter ö.
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Examples : Venn Diagrams
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Exclusive Events:
Two events are said to be mutually exclusive if they can
not occur simultaneously.
If A and B are two mutually exclusive events then
Probability Rules
4.
If A and B are mutually exclusive then
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Conditional Probability
Let A and B be two events, and suppose that the event B
has already occurred. What is the probability of the
occurrence of the event A in this situation?
In this case we write
given B.
, read the probability of a A
The conditional probability
1.
is given by
The success rate of two hospitals in performing open-heart surgery is being evaluated. The
following data represent a sample of open-heart surgery patients obtained over a period of
15 years for each of the hospitals:
Number of open-heart
surgeries performed
Number of patients alive after
one year of having the surgery
Hospital 1
1,500
1,446
Hospital 2
1,800
1,750
a)
Describe the population.
b)
Describe the sample.
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2.
3.
Describe the following as statistical or probabilistic
a)
The closing of the price of General Electric next trading day.
b)
The likelihood of wining the lottery.
Identify the number as either continuous or discrete.
a)
The height of 2-year-old maple tree is 28 ft.
b)
The number of tax payers that will receive a refund in tax year 2005.
4.
State what is wrong with the claim in this add.
An airline company advertises that 100% of their flights are on time after checking 5
randomly selected flights and finding that these 5 were on time.
5.
Determine the following type of study .
A sample of fish is taken from a lake to measure the effect of pollution from a nearby
factory on the fish.
A) Experimental
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B) Observational
Determine the type of sampling
A sample consists of every 49th student from a group of 496 students.
A) Cluster
B) Random
C) Systematic
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D) Stratified E) Convenience
Example
The following table gives political affiliation of a
group of 500 men and women
Men
Wome
n
Total
Republican
140
120
260
Democrat
90
150
240
Total
230
270
500
What is the probability a person selected at random is
a)
a Republican?
b) a Republican given the person is a women?
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The Multiplication Rule
Independent Events
Two events are said to be independent if the occurrence of
one event does not change the probability of the
occurrence of the other. If two events are not independent,
we say they are dependent.
If two events A and B are independent, we have:
The Multiplication Rule for independent events is given by
Note:
If events A and B are independent, then
The converse is also true.
Example
In the preceding example, are the event “ Women” and the
event “Republican” are independent?
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Tree Diagrams:
A fair coin is tossed and a fair die is rolled.
Find the all possible outcomes.
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