Axioms, Interpretations and Properties

Given an experiment and a sample space S, the
objective of probability is to assign to each event
A a number P(A) that gives a precise measure of
the chance that A will occur.

To ensure that these assignments will be
consistent with intuitive notions of probability,
all assignments should satisfy the following
basic axioms .
2

For any event A, P( A)  0 .

P(S)=1

If A1 , A2 ,
events,
is an infinite collection of disjoint
P( A1  A2 
)   i 1 P( Ai )

3

P   0

Proof: Consider A1  , A2  , , disjoint sets
A   . Then by axiom 3,
with
i i
P      P    , which implies that P     0 .

Also if A1 , A2 , , Ak are disjoint, append
Ak 1  , Ak  2  , Then
P

k
i 1
 
Ai  P

i 1

Ai  i 1 P  Ai   i 1 P  Ai 

k
4


For an event A , P  A  P  A   1 , and thus
P  A   1  P  A .
Proof: Since A  A  S and the sets are
disjoint, 1  P( S )  P  A  A   P  A   P  A  .
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

For any event A , P  A  1 .
Proof: 1  P  A  A   P  A  P  A   P  A 
by Axiom 1.
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
For any events A and B ,
P  A  B   P  A  P  B   P  A  B 

Proof: Since A  B  A   B  A  (which are
disjoint sets),
P  A  B   P  A  P  B  A 
 P  A   P  B   P  A  B  
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
In a certain residential suburb, 60% of all
households get Internet service from the local
cable company, 80% get television from that
company, and 50% get both. What is the
probability that a randomly selected household
gets at least one of the two services from the
company, and what is the probability that they
get exactly one?

The formula for the probability of the union of
events is relevant, and the solutions become
easy if we use a Venn diagram.
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 For any events A, B, and C,
P  A  B  C   P  A  P  B   P  C   P  A  B 
P  A  C   P  B  C   P  A  B  C 
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
Let Ei denote the simple events of a sample
space each consisting of a single outcome.
Then the probability of a compound event A
consisting of the simple events Ei is:
P  A   E ' s in A P  Ei 
i
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
In some experiments consisting of N
outcomes, it is reasonable to assign equal
probability to all N simple events. Then
1   i 1 P  Ei    i 1 p  pN
N
N
1
 Thus p 
.
N
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
In an experiment that can be repeated, let
n(A) be the number of occurrences of event A
and n the number of repetitions of the
experiment. Then n(A)/n is called the relative
frequency of the event A.

As n gets large, this relative frequency
converges to a limiting value that we identify
with P(A).
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
In some instances we make probabilistic
statements about situations that are not
repeatable. As an example: “It is likely that
our company will be awarded the contract.”

Any assignment of probability in this case is
subjective because different observers would
have different prior information and different
opinions on the likelihood.
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