Sample spaces and events

An experiment is an activity or process whose
outcome is subject to uncertainty

The sample space of an experiment, denoted
by S , is the set of all possible outcomes.
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
If one observes the gender of the next child
born in the local hospital, S  M , F  (two
possible outcomes)

If we examine three fuses and record whether
each is non-defective or defective,
S   NNN , NND, NDN , NDD, DDD, DDN , DND, DNN 
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
If two gas stations at a certain intersection each
have six pumps and the experiment is to record
how many pumps are in use, the 49 possible
outcomes are the ordered pairs
S   0,0  ,  0,1 , ,  0,6  , 1,0  , 1,1 , , 1,6  , ,
 6,0  ,
,  6,6 
where  i, j  gives the number of pumps in use at
stations i and j .
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
If the experiment consists of selecting and
compiling C++ programs until one compiles
on the first run, the sample space may be
written as Y , NY , NNY , NNNY ,  , where
N stands for a failure, and Y for a success.
Here there are a (countably) infinite number of
possible outcomes.
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
An event is any collection (subset) of
outcomes contained in the sample space S .

An event is simple if it consists of exactly one
outcome and compound if it consists of more
than one outcome.
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
In the gas pump example, events include:
-- The number of pumps in use is the same for
both stations
A   0,0  , 1,1 ,  2, 2  ,  3,3 ,  4, 4  , 5,5  ,  6,6 
-- The total number of pumps in use is four
B   0, 4  , 1,3 ,  2, 2  ,  3,1 ,  4,0 
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
Events include:
-- A  Y , NY , NNY  the event that at most
three programs are examined
-- B  Y , NNY , NNNNY , NNNNNNY  , the event
that ?
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
An event is a set, so relationships and
results from elementary set theory can
be used to study events.
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
The complement of an event A , denoted by A
is the set of all outcomes in S that are not in A.
The union of two events A and B , denoted by A  B
is the set of all outcomes in either A or B or in both.


The intersection of two events A and B, ( A  B ) is
the set of all outcomes that are in both A and B.
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
A pictorial representation of events and
manipulations with events is obtained by
using Venn diagrams.

To construct a Venn diagram, draw a
rectangle whose interior represents the
sample space S.
Then any event is
represented as the interior of a closed circle.
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

In the program compilation experiment,
define A  Y , NY , NNY  , B  Y , NNY , NNNNY .
Then A   NNNY , NNNNY , NNNNNY , 
A  B  Y , NY , NNY , NNNNY 
A  B  Y , NNY 
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
Union and intersection of sets can be
extended to more than two sets
 A B C
is the set of outcomes in at least
one of the three events
 A B C
is the set of outcomes in all three
events
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
Let  denote the empty or null event (the
event consisting of no outcomes). When
A  B   , A and B are said to be mutually
exclusive or disjoint events.
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
A small city has three automobile
dealerships: a GM dealer selling Chevrolets
and Buicks, a Ford dealer selling Fords and
Lincolns, and a Toyota dealer.
If an
experiment consists of observing the brand of
the next car sold, then the events
A  Chevrolet, Buick
B  Ford,Lincoln
are mutually exclusive.
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