Before considering the methodology for estimating system reliability,
some basic concepts from probability theory should be reviewed.
The terms that follow are important in creating and analyzing reliability
block diagrams.
Experiment (E):
An experiment is any well-defined action that may result in a number of
outcomes. For example, the rolling of dice can be considered an experiment.
Outcome (O):
An outcome is defined as any possible result of an experiment.
Sample space (S):
The sample space is defined as the set of all possible outcomes of an
experiment.
Event: An event is a collection
of outcomes.
Union of two events A
and B (AB): The
union of two events A
and Bis the set of outcomes that belong
to A or B
or both.
Intersection of two events A and B
(AB): The intersection of two
events A and B
is the set of outcomes that belong to both A
and B.
Complement of event A
(): A complement of an event A
contains all outcomes of the sample space, S,
that do not belong to A.
Null event ():
A null event is an empty set and it has no outcomes.
Probability: Probability is a numerical measure of
the likelihood of an event relative to a set of alternative events. For
example, there is a 50% probability of observing heads relative to observing
tails when flipping a coin (assuming a fair or unbiased coin).
Experiment Example
Consider an experiment that consists of the rolling of a six-sided die.
The numbers on each side of the die are the possible outcomes. Accordingly,
the sample space is S
= {1, 2, 3, 4, 5, 6}.
Let A be the event of
rolling a 3, 4 or 6 (A
= {3, 4, 6}) and let B
be the event of rolling a 2, 3 or 5 (B
= {2, 3, 5,}).
The union of A
and B is: AB = {2,
3, 4, 5, 6}
The intersection of A
and B is: AB = {3}.
The complement of A
is: = {1, 2, 5}.
Probability Properties, Theorems and Axioms
The probability of an event A
is expressed as P(A), and has the following properties:
0 P(A) 1.
P(A) = 1 - P()
P() = 0.
P(S) = 1.
In other words, when an event is certain to occur, it has a probability
equal to 1; when it is impossible for the event to occur, it has a probability
equal to 0.
It can also be shown that the probability of the union of two events
A and B
is:
(1)
Similarly, the probability of the union of three events, A,
B and C
is given by:
Mutually Exclusive Events
Two events A and B are said to be mutually exclusive
if it is impossible for them to occur simultaneously (AB = C). In such cases, the expression
for the union of these two events reduces to the following, since the
probability of the intersection of these events is defined as zero.
Conditional Probability
The conditional probability of two events A
and B is defined as the
probability of one of the events occurring, knowing that the other event
has already occurred. The expression below denotes the probability of
A occurring given that
B has already occurred.
(2)
Note that knowing that event B
has occurred reduces the sample space.
Independent Events
If knowing B gives no
information about A, then
the events are said to be independent
and the conditional probability expression reduces to:
(3)
From the definition of conditional probability, Eqn. (2) can be written
as:
(4)
Since events A and B are independent, the expression
reduces to:
(5)
If a group of n events
Ai
are independent, then:
(6)
As an illustration, consider the outcome of a six-sided die roll. The
probability of rolling a 3 is one out of six or:
All subsequent rolls of the die are independent events, since knowing
the outcome of the first die roll gives no information as to the outcome
of subsequent die rolls (unless the die is loaded). Thus the probability
of rolling a 3 on the second die roll is again:
However, if one were to ask the probability of rolling a double 3 with
two dice, the result would be:
Statistical Background Example 1
Consider a system, as shown in Figure 3.1, where two hinged members
are holding a load in place.
Figure 3.1: System
for Example 1.
The system fails if either member fails and the load is moved from its
position.
Let A
event of failure of Component 1 and let = the
event of not failure of Component 1.
Let B
event of failure of Component 2 and let = the event of
not failure of Component 2.
Failure occurs if Component 1 or Component 2 or both fail. The system
probability of failure (or unreliability) is:
Assuming independence (or that the failure of either component is not
influenced by the success or failure of the other component), the system
probability of failure becomes the sum of the probabilities of A
and B occurring minus
the product of the probabilities:
Another approach is to calculate the probability of the system not failing,
or the reliability of the system:
Then, the probability of system failure is simply 1 (or 100%) minus
the reliability:
Statistical Background Example 2
Consider a system of a load being held in place by two rigid members,
as shown in Figure 3.2.
Figure 3.2: System
for Example 2.
Let A
event of failure of Component 1.
Let B
event of failure of Component 2.
The system fails if Component 1 fails and Component
2 fails. In other words, both components must fail for the system to fail.
The system probability of failure is defined as the intersection of
events A and B:
(7)
Case 1
Assuming independence (i.e.
either one of the members is sufficiently strong to hold the load in place),
the probability of system failure becomes the product of the probabilities
of A and B
failing:
The reliability of the system now becomes:
(8)
Case 2
If independence is not assumed (e.g.
when one component fails the other one is then more likely to fail), then
the simplification given in Eqn. (8) is no longer applicable. In this
case, Eqn. (7) must be used. We will examine this dependency in later
sections under the subject of load
sharing.
B): The
union of two events A
and B is the set of outcomes that belong
to A or B
or both.
B): The intersection of two
events A and B
is the set of outcomes that belong to both A
and B.
): A complement of an event A
contains all outcomes of the sample space, S,
that do not belong to A.
):
A null event is an empty set and it has no outcomes.
(1)
(2)
(3)
(4)
(5)
(6)
= the event of
not failure of Component 2.
(7)
(8)