Axiomatic Probability
The objective of probability is to assign to each event A a number
P(A), called the probability of the event A, which will give a precise
measure of the chance thtat A will occur.
Probability Axioms:
AXIOM 1 For any event A, P(A) ≥ 0.
AXIOM 2 P(S) = 1.
AXIOM 3 If A1 , A2 , A3 , . . . is an infinite collection
of disjoint events,
P∞
then P(A1 ∪ A2 ∪ A3 ∪ · · · ) = i=1 P(Ai )
Liang Zhang (UofU)
Applied Statistics I
June 10, 2008
1/8
Axiomatic Probability
Proposition
P(∅) = 0 where ∅ is the null event. This in turn implies that the property
contained in Axiom 3 is valid for finite collection of events, i.e. if
A1 , A2 , . . . , An is a finite collection
of disjoint events, then
Pn
P(A1 ∪ A2 ∪ · · · ∪ A3 ) = i=1 P(Ai )
Liang Zhang (UofU)
Applied Statistics I
June 10, 2008
2/8
Axiomatic Probability
Examples:
1. Consider the coin tossing experiment and we are only interested in
tossing the coin one time. Then S = {H, T}.
Since P(S) = 1 (Axiom 1), and the event {H} and {T} are mutually
disjoint, by Axiom 3, we have
P({H}) + P({T }) = P({H} ∪ {T }) = P(S) = 1
If the coin is fair, we should assign 0.5 to P({H}) and 0.5 to P({T }).
If the coin is more likely to give a Head, then 0.8 for P({H}) and 0.2 for
P({T }) may be suitable.
In fact, if p is any fixed number between 0 and 1, then P({H}) = p , and
P({T }) = 1 − p is an assignment consistent with the axioms.
Liang Zhang (UofU)
Applied Statistics I
June 10, 2008
3/8
Axiomatic Probability
Examples:
2. Consider again the coin tossing example. However, this time we are
interested in getting a Head, i.e. we toss a coin many times untill we get a
Head. Then S = {H, TH, TTH, TTTH, TTTTH, . . . }.
If P({H}) = 0.4 then P({T }) = 0.6, P({TH}) = (0.4)0.6,
P({TTH}) = (0.4)(0.6)2 , P({TTTH}) = (0.4)(0.6)3 , . . . .
Since {H}, {TH}, {TTH}, {TTTH}, {TTTTH}, . . . are mutually disjoint
and S = {H} ∪ {TH} ∪ {TTH} ∪ {TTTH} ∪ {TTTTH} ∪ . . . , we have
1 = 0.4 + (0.4)(0.6) + (0.4)(0.6)2 + (0.4)(0.6)3 + · · ·
Liang Zhang (UofU)
Applied Statistics I
June 10, 2008
4/8
Axiomatic Probability
More Probability Properties
Proposition
For any event A, P(A) + P(A0 ) = 1, from which P(A) = 1 − P(A0 ).
Example 2.13
Consider a system of five identical components connected in series, as
illustrated below.
Denote a component failure by F and success by S. Let A be the event
that the system fails. For A to occur, at least one of the individual
components must fail. If we know P({F }) = 0.1, then what is P(A)?
Liang Zhang (UofU)
Applied Statistics I
June 10, 2008
5/8
Axiomatic Probability
Proposition
For any event A, P(A) ≤ 1 .
Proposition
For any two events A and B,
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
A Venn Diagram proof:
=
Liang Zhang (UofU)
+
Applied Statistics I
June 10, 2008
6/8
Axiomatic Probability
Example 2.14
In a certain residential suburb, 60% of all households subscribe to the
metropolitan newspaper published in a nearby city, 80% subscribe to the
local paper, and 50% of all households subscribe to both papers. If a
househlld is selected at random, what is the probability that it subscribes
to (1)at least one of the two newspapers and (2) exactly one of the two
newspapers?
Liang Zhang (UofU)
Applied Statistics I
June 10, 2008
7/8
Axiomatic Probability
Proposition
For any three events A, B, and C ,
P(A ∪ B ∪ C ) =P(A) + P(B) + P(C )
− P(A ∩ B) − P(B ∩ C ) − P(C ∩ A)
+ P(A ∩ B ∩ C )
A Venn Diagram interpretation:
Liang Zhang (UofU)
Applied Statistics I
June 10, 2008
8/8