24 2. FUNDAMENTAL RELIABILITY MATHEMATICS
An event refers to a single outcome or a group of outcomes of an experiment. For example, the
even number of rolling a die, that is, f2,4,6gis an event, which includes three possible outcomes.
e number 4 of rolling die is also an event, which has only one outcome. For another example,
that the yield strength is more than 32 ksi for a material tensile test is an event, which includes
infinite possible outcomes.
Sample space is defined as the event that includes all the possible outcomes of an experiment.
For example, the sample space of rolling a die consists of numbers 1, 2, 3, 4, 5, and 6. e
sample space of tossing a coin will consist of “head” and “tail.” e sample space of a material
tensile test experiment for yield strength will consist of the infinite possible outcomes, that is,
fyield strength 0g.
Mutually exclusive events are the events that cannot happen at the same time. For example, the
event of the tail and the event of the head in tossing a coin are mutually exclusive. e event of
a failed component and the event of a safe component are mutually exclusive events too because
a component cannot be a safe status or a failure status at the same time. For another example,
the event of a number less than 3 and the event of even number in rolling die are not mutually
exclusive events. If number 2 in an experiment of rolling a die happens, both events happen.
2.3 SET THEORY
e set theory can conveniently describe the operation on the outcomes or events of an exper-
iment and will help us smoothly to understand the concepts and related simple operations of
probability.
A set, denoted by a capital letter such as A, is a well-defined collection of objects so that for
any given object we can say whether or not it belongs to the set. A set is the equivalent term
of the event. In this book, the object means sample points or outcomes of an experiment. A
set can be expressed by braces containing the specified sample point or simple points. For
example, in rolling a die experiment, a set containing numbers 1 and 4 can be expressed as
A D
f
1; 4
g
, where the capital symbol A is the name of this set and
f
1; 4
g
represents the collec-
tion of sample point 1 and sample point 4. For an experiment of ultimate tensile strength S
u
,
B D
f
5 ksi < S
u
< 20 ksi
g
is a set. e set B includes the sample point of the ultimate tensile
strength is larger than 5 ksi and less than 20 ksi.
A universal set, denoted by Greek letter , is a collection of all possible sample points of the
experiment. For example, the universal set of rolling dice is D
f
1; 2; 3; 4; 5; 6
g
. For another
example, the universal set of the status of a component is D
f
safe; failure
g
.
An empty set or null set, denoted by the symbol ;, is a set containing no sample point of the
experiment. A null set can be expressed by ; D
f g
. e creation of a null set is mainly for the
set operations.