# Probability Tree Analysis

This method can be used for performing reliability-related task analysis by diagram- matically representing human actions and other associated events in question. In this case, diagrammatic task analysis is represented by the branches of the probability tree. More specifically, the tree’s branching limbs represent each event’s outcome (i.e., success or failure) and each branch is assigned probability of occurrence [17].

Some of the advantages of this method are flexibility for incorporating (i.e., with some modifications) factors such as interaction effects, emotional stress, and interaction stress; simplified mathematical computations; and a visibility tool. It is to be noted that the method can also be used for evaluating reliability of networks such as series, parallel, and series-parallel. The method’s application to such configurations is demonstrated in Dhillon [18].

Nonetheless, the following example demonstrates the application of this method:

Example 4.8

Assume that a person has to perform two independent and distinct tasks m and n to operate an engineering system. Task in is performed before task n. Furthermore, each of these two tasks can be conducted either correctly or incorrectly.

Develop a probability tree and obtain an equation for probability of not successfully accomplishing the overall mission (i.e., not operating the engineering system correctly) by the person.

In this example, the person first performs task m correctly or incorrectly and then proceeds to perform task n. This task can also be performed either correctly or incorrectly. Figure 4.11 depicts a probability tree for the entire scenario.

The symbols used in the figure are defined below.

m denotes the event that task m is performed correctly. m denotes the event that task m is performed incorrectly.

FIGURE 4.11 A probability tree for performing tasks in and n.

n denotes the event that task n is performed correctly. n denotes the event that task n is performed incorrectly.

In Figure 4.11, the term nm denotes operating the engineering system successfully (i.e., overall mission success). Thus, the probability of occurrence of events inn is given by [ 18]

where

Pm is the probability of performing task m correctly.

P„ is the probability of performing task n correctly.

Similarly, in Figure 4.11, the terms mn, inn, and inn denote three distinct possibilities of not operating the engineering system correctly or successfully. Thus, the probability of not successfully accomplishing the overall mission (i.e., not operating the engineering system correctly) by the person is

where

p_ is the probability of performing task n incorrectly. p_ is the probability of performing task m incorrectly.

1 m

Pf is the probability of not successfully accomplishing the overall mission (i.e., mission failure).

Example 4.9

Assume that in Example 4.8, the probabilities of the person not performing tasks m and n correctly are 0.3 and 0.25, respectively. Calculate the probability of correctly operating the engineering system by the person.

Thus, we have P = 0.3 and P = 0.25.

m n

Since P +Pm = 1 and P +Pn = 1, we have

m n

By substituting Equations (4.35)-(4.36) and the specified data values into Equation (4.33), we obtain

Thus, probability of correctly operating the engineering system (i.e., the probability of occurrence of events mri) by the person is 0.525.

# Binomial Method

This method is used for evaluating the reliability of relatively simple systems, such as series and parallel systems/networks. For such systems’/networks’ reliability evaluation, this is one of the simplest methods. However, in the case of complex systems/ networks the method becomes a trying task. The method can be applied to systems/ networks with independent identical or non-identical units. The following formula is the basis for the method [16]:

where

n is the number of non-identical units/components.

R, is the /th unit reliability.

Fj is the /th unit failure probability.

Example 4.10

Using Equation (4.37) develop reliability expressions for parallel and series networks having two non-identical and independent units each.

In this case, since n = 2 from Equation (4.37) we get

where

Rx is the reliability of unit l.

R-, is the reliability of unit 2.

F1 is the failure probability of unit 1.

F-, is the failure probability of unit 2.

Thus, using Equation (4.38), we write the following reliability expression for the parallel network having two non-identical units:

where

Rp2ls thetwo non-identical units parallel network reliability.

Since (+ Fj) = 1 and (R2 + F2) = 1, Equation (4.39) becomes

By rearranging Equation (4.40) we obtain

Finally, the two non-identical units series network reliability from Equation (4.38) is

where

Rsl is the two non-identical units series network reliability.

Thus, reliability expressions for parallel and series networks having two nonidentical and independent units each are given by Equations (4.41) and (4.42), respectively.

# Problems

• 1. Describe failure modes and effect analysis method and its advantages.
• 2. What are the main objectives of conducting FTA and the main prerequisites associated with FTA?
• 3. Assume that a windowless room has five light bulbs and one switch. Develop a fault tree for the undesired (i.e., top) fault event, Dark room, if the switch only fails to close.
• 4. What are the assumptions associated with the Markov method?
• 5. Assume that the constant failure and repair rates of an engineering system are 0.0005 failures per hour and 0.0007 repairs per hour, respectively. Calculate the engineering system steady-state unavailability and unavailability during a 50-hour mission.
• 6. Assume that five independent and identical units form a bridge network and reliability of each unit is 0.8. Calculate the network reliability by using the delta-star method and the decomposition approach. Compare both the results.
• 7. Assume that a person has to perform three independent and distinct tasks a, b, and c to operate an engineering system. Task a is performed before task b, and task b before task c. Furthermore, each of these tasks can be performed either correctly or incorrectly. Develop a probability tree and obtain an expression for probability of not successfully accomplishing the overall mission (i.e., not operating the engineering system correctly) by the person.
• 8. Compare probability tree analysis with FTA.
• 9. Describe binomial method and write down its basic formula.
• 10. Develop reliability expression for a parallel network with three independent and non-identical units by using the binomial method.

# References

• 1. Grant Ireson, W., Coombs, C.F.. Moss, R.Y., Editors, Handbook of Reliability Engineering and Management, McGraw-Hill, New York, 1996.
• 2. Dhillon, B.S., Design Reliability: Fundamentals and Applications, CRC Press, Boca Raton, Florida. 1999.
• 3. RDG-376, Reliability Design Handbook, Reliability Analysis Center, Rome Air Development Center, Griffis Air Force Base, Rome, New York, 1976.
• 4. AMCP 706-196, Engineering Design Handbook: Development Guide for Reliability, Part II: Design for Reliability, US Army Material Command (AMC), Washington, DC, 1976.
• 5. Dhillon, B.S., Proctor, C.L., Reliability Analysis of Multistate Device Networks, Proceedings of the Annual Reliability and Maintainability Symposium, 1976, pp. 31-35.
• 6. Jordan, W.E., Failure Modes, Effects, and Criticality Analyses, Proceedings of the Annual Reliability and Maintainability Symposium, 1972, pp. 30-37.
• 7. Omdahl, T.P., Editor, Reliability, Availability’, and Maintainability (RAM) Dictionary, American Society for Quality Control (ASQC) Press, Milwaukee, Wisconsin, 1988.
• 8. MIL-F-18372 (Aer), General Specification for Design, Installation, and Test of Aircraft Flight Control Systems, Bureau of Naval Weapons, Department of the Navy, Washington. DC.
• 9. Palady, P, Failure Modes and Effects Analysis, PT Publications, West Palm Beach, Florida, 1995.
• 10. McDermott, R.E.. Mikulak. K.J., Beauregard. M.R., The Basics of FMEA, Quality Resources, New York, 1996.
• 11. Dhillon, B.S., Singh, C., Engineering Reliability: New Techniques and Applications, John Wiley, New York. 1981.
• 12. Schroder, R.J., Fault Tree for Reliability Analysis, Proceedings of the Annual Symposium on Reliability, 1970, pp. 170-174.
• 13. Mears, P, Quality Improvement Tools and Techniques, McGraw Hill, New York, 1995.
• 14. Shooman, M.L., Probabilistic Reliability: An Engineering Approach, McGraw Hill, New' York, 1968.
• 15. Dhillon, B.S., The Analysis of the Reliability’ of Multi-State Device Networks, Ph.D. Dissertation, 1975. Available from the National Library' of Canada, Ottaw'a.
• 16. Dhillon, B.S., Reliability in Systems Design and Operation, Van Nostrand Reinhold, New' York, 1983.
• 17. Swain, A.D., A Method for Peiforming a Human Factors Reliability’ Analysis, Report No. SCR-685, Sandia Corporation, Albuquerque, New Mexico, USA, August 1963.
• 18. Dhillon, B.S., Human Reliability’: With Human Factors, Pergamon Press, New York, 1986.