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Summary Notes  Probability Theory
Basic De…nitions
De…nition 1
The term
probability
refers to the study of randomness and uncertainty. In any
situation in which one of a number of possible outcomes may occur, the theory of probability
provides methods for quantifying the chances, or likelihoods, associated with the various
outcomes.
De…nition 2
An
experiment
is any action or process that generates observations.
De…nition 3
The
sample space
of an experiment, denoted by
S
, is the set of all possible outcomes
of the experiment.
De…nition 4
An
event
is any collection (subset) of outcomes contained in the sample space
S
. An
event is said to be
simple
if it consists of exactly one outcome and
compound
if it consists
of more than one outcome.
De…nition 5
The
union
of two events
A
and
B
, denoted by
A
[
B
and read ”
A
or
B
,” is the
event consisting of all outcomes that are either in
A
or in
B
or in both events (so that the
union includes outcomes for which both
A
and
B
occur as well as outcomes for which exactly
one occurs).
De…nition 6
The
intersection
of two events
A
and
B
, denoted by
A
\
B
(sometimes
AB
) and
read ”
A
and
B
,” is the event consisting of all outcomes that are in both
A
and
B
.
De…nition 7
The
complement
of an event
A
, denoted by
A
0
(sometimes
A
c
or
A
), is the set of
all outcomes in
S
that are not contained in
A
.
De…nition 8
When
A
and
B
have no outcomes in common, they are said to be
mutually exclu
sive
or
disjoint
events.
Elementary Laws from Set Theory
Commutative Laws
E
[
F
=
F
[
E
;
E
\
F
=
F
\
E
.
Associative Laws
E
[
(
F
[
G
) = (
E
[
F
)
[
G
;
E
\
(
F
\
G
) = (
E
\
F
)
\
G
.
Distributive Laws
(
E
\
F
)
[
G
= (
E
[
G
)
\
(
F
[
G
)
;
(
E
[
F
)
\
G
= (
E
\
G
)
[
(
F
\
G
)
DeMorgan’s First Law
Ã
n
[
i
=1
E
i
!
0
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 Fall '10
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