Basic Definitions

Before considering the methodology for estimating system reliability, some basic concepts from probability theory must be reviewed.

The terms that follow are important in creating and analyzing reliability block diagrams.

1. 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.

2. Outcome (O): An outcome is defined as any possible result of an experiment.

3. Sample space (S): The sample space is defined as the set of all possible outcomes of an experiment.

4. Event: An event is a collection of outcomes.

5. Union of two events A and B (A B): The union of two events A and B is the set of outcomes that belong to A or B or both.

6. Intersection of two events A and B (A B): The intersection of two events A and B is the set of outcomes that belong to both A and B.

7. Complement of event A (): A complement of an event A contains all outcomes of the sample space, S, that do not belong to A.

8. Null event (): A null event is an empty set and it has no outcomes.

9. 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,}).

1. The union of A and B is: A B = {2, 3, 4, 5, 6}

2. The intersection of A and B is: A B = {3}.

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:

1. 0 P(A) 1.

2. P(A) = 1 - P()

3. P() = 0.

4. P(S) = 1.

In other words, when an event is certain to occur, it has a probability equal to 1; when it is impossible 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 (A B = 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 are independent, then:

(6)

As an illustration, consider the outcome of a six-sided dice roll. The probability of rolling a 3 is one out of six or:

All subsequent rolls of the dice are independent events, since knowing the outcome of the first dice roll gives no information as to the outcome of subsequent dice rolls (unless the dice are loaded). Thus the probability of rolling a 3 on the second dice 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.

1. Let A event of failure of Component 1 and let = the event of not failure of Component 1.

2. 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